arXiv subjects

Salvatore Torquato

Publications and source records attributed to Salvatore Torquato.

At least 127 records · Page 7Linked to original sources

arXiv2014

Diagnosing hyperuniformity in two-dimensional disordered jammed-packings of soft spheres

Hyperuniformity characterizes a state of matter for which density fluctuations diminish towards zero at the largest length scales. However, the task of determining whether or not an experimental system is hyperuniform is experimentally challenging due to finite-resolution, noise and sample-size effects that influence characterization measurements. Here we explore these issues, employing video optical microscopy to study hyperuniformity phenomena in disordered two-dimensional jammed packings of soft spheres. Using a combination of experiment and simulation we characterize the detrimental effects of particle polydispersity, image noise, and finite-size effects on the assignment of hyperuniformity, and we develop a methodology that permits improved diagnosis of hyperuniformity from real-space measurements. The key to this improvement is a simple packing reconstruction algorithm that incorporates particle polydispersity to minimize free volume. In addition, simulations show that hyperuniformity can be ascertained more accurately in direct space than in reciprocal space as a result of finite sample-size. Finally, experimental colloidal packings of soft polymeric spheres are shown to be hyperuniform.

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arXiv2014

Equilibrium Phase Behavior and Maximally Random Jammed State of Truncated Tetrahedra

Systems of hard nonspherical particles exhibit a variety of stable phases with different degrees of translational and orientational order, including isotropic liquid, solid crystal, rotator and a variety of liquid crystal phases. In this paper, we employ a Monte Carlo implementation of the adaptive-shrinking-cell (ASC) numerical scheme and free-energy calculations to ascertain with high precision the equilibrium phase behavior of systems of congruent Archimedean truncated tetrahedra over the entire range of possible densities up to the maximal nearly space-filling density. In particular, we find that the system undergoes two first-order phase transitions as the density increases: first a liquid-solid transition and then a solid-solid transition. The isotropic liquid phase coexists with the Conway-Torquato (CT) crystal phase at intermediate densities. At higher densities, we find that the CT phase undergoes another first-order phase transition to one associated with the densest-known crystal. We find no evidence for stable rotator (or plastic) or nematic phases. We also generate the maximally random jammed (MRJ) packings of truncated tetrahedra, which may be regarded to be the glassy end state of a rapid compression of the liquid. We find that such MRJ packings are hyperuniform with an average packing fraction of 0.770, which is considerably larger than the corresponding value for identical spheres (about 0.64). We conclude with some simple observations concerning what types of phase transitions might be expected in general hard-particle systems based on the particle shape and which would be good glass formers.

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arXiv2014

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

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 optimization scheme [Nature 460, 876 (2009)] to find maximally dense particle packings in d-dimensional Euclidean space $\mathbb{R}^d$. While the original implementation was designed to study spheres and convex polyhedra in $d\ge3$, our implementation focuses on $d=2$ and extends the algorithm to include both concave polygons and certain convex or concave non-polygonal 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 "moon-like" 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 non-lattice 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 moon-like 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.

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arXiv2014

Dense periodic packings of tori

Dense packings of nonoverlapping bodies in three-dimensional Euclidean space are useful models of the structure of a variety of many-particle systems that arise in the physical and biological sciences. Here we investigate the packing behavior of congruent ring tori, which are multiply connected nonconvex bodies of genus 1, as well as horn and spindle tori. We analytically construct a family of dense periodic packings of unlinked tori guided by the organizing principles originally devised for simply connected solid bodies [Torquato and Jiao, PRE 86, 011102 (2012)]. We find that the horn tori as well as certain spindle and ring tori can achieve a packing density higher than the densest known packing of both sphere and ellipsoids. In addition, we study dense packings of cluster of pair-linked ring tori (i.e., Hopf links).

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arXiv2014

Avian photoreceptor patterns represent a disordered hyperuniform solution to a multiscale packing problem

Optimal spatial sampling of light rigorously requires that identical photoreceptors be arranged in perfectly regular arrays in two dimensions. Examples of such perfect arrays in nature include the compound eyes of insects and the nearly crystalline photoreceptor patterns of some fish and reptiles. Birds are highly visual animals with five different cone photoreceptor subtypes, yet their photoreceptor patterns are not perfectly regular. By analyzing the chicken cone photoreceptor system consisting of five different cell types using a variety of sensitive microstructural descriptors, we find that the disordered photoreceptor patterns are ``hyperuniform'' (exhibiting vanishing infinite-wavelength density fluctuations), a property that had heretofore been identified in a unique subset of physical systems, but had never been observed in any living organism. Remarkably, the photoreceptor patterns of both the total population and the individual cell types are simultaneously hyperuniform. We term such patterns ``multi-hyperuniform'' because multiple distinct subsets of the overall point pattern are themselves hyperuniform. We have devised a unique multiscale cell packing model in two dimensions that suggests that photoreceptor types interact with both short- and long-ranged repulsive forces and that the resultant competition between the types gives rise to the aforementioned singular spatial features characterizing the system, including multi-hyperuniformity.

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arXiv2014

Jammed lattice sphere packings

We generate and study an ensemble of isostatic jammed hard-sphere lattices. These lattices are obtained by compression of a periodic system with an adaptive unit cell containing a single sphere until the point of mechanical stability. We present detailed numerical data about the densities, pair correlations, force distributions, and structure factors of such lattices. We show that this model retains many of the crucial structural features of the classical hard-sphere model and propose it as a model for the jamming and glass transitions that enables exploration of much higher dimensions than are usually accessible.

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arXiv2013

Photonic band gap in isotropic hyperuniform disordered solids with low dielectric contrast

We report the first experimental demonstration of a TE-polarization photonic band gap (PBG) in a 2D isotropic hyperuniform disordered solid (HUDS) made of dielectric media with a index contrast of 1.6:1, very low for PBG formation. The solid is composed of a connected network of dielectric walls enclosing air-filled cells. Direct comparison with photonic crystals and quasicrystals permitted us to investigate band-gap properties as a function of increasing rotational isotropy. We present results from numerical simulations proving that the PBG observed experimentally for HUDS at low index contrast has zero density of states. The PBG is associated with the energy difference between complementary resonant modes above and below the gap, with the field predominantly concentrated in the air or in the dielectric. The intrinsic isotropy of HUDS may offer unprecedented flexibilities and freedom in applications (i. e. defect architecture design) not limited by crystalline symmetries.

cond-mat.mes-hall↗

arXiv2013

Isotropic Band Gaps and Freeform Waveguides Observed in Hyperuniform Disordered Photonic Solids

Recently, disordered photonic media and random textured surfaces have attracted increasing attention as strong light diffusers with broadband and wide-angle properties. We report the first experimental realization of an isotropic complete photonic band gap (PBG) in a two-dimensional (2D) disordered dielectric structure. This structure is designed by a constrained-optimization method, which combines advantages of both isotropy due to disorder and controlled scattering properties due to low density fluctuations (hyperuniformity) and uniform local topology. Our experiments use a modular design composed of Al2O3 walls and cylinders arranged in a hyperuniform disordered network. We observe a complete PBG in the microwave region, in good agreement with theoretical simulations, and show that the intrinsic isotropy of this novel class of PBG materials enables remarkable design freedom, including the realization of waveguides with arbitrary bending angles impossible in photonic crystals. This first experimental verification of a complete PBG and realization of functional defects in this new class of materials demonstrates their potential as building blocks for precise manipulation of photons in planar optical micro-circuits and has implications for disordered acoustic and electronic bandgap materials.

cond-mat.mes-hall↗

arXiv2013

Optical cavities and waveguides in hyperuniform disordered photonic solids

Using finite difference time domain and band structure computer simulations, we show that it is possible to construct optical cavities and waveguide architectures in hyperuniform disordered photonic solids that are unattainable in photonic crystals. The cavity modes can be classified according to the symmetry (monopole, dipole, quadrupole,etc.) of the confined electromagnetic wave pattern. Owing to the isotropy of the band gaps characteristic of hyperuniform disordered solids, high-quality waveguides with freeform geometries (e.g., arbitrary bending angles) can be constructed that have no analogue in periodic or quasiperiodic solids. These capabilities have implications for many photonic applications.

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arXiv2013

Extreme lattices: symmetries and decorrelations

We study statistical and structural properties of extreme lattices, which are the local minima in the density landscape of lattice sphere packings, in $d$-dimensional Euclidean space $\mathbb{R}^d$. Specifically, we ascertain the distributions of densities, kissing numbers and numbers of symmetries of the packings across a wide range of dimensions using the stochastic Voronoi algorithm. The degree to which the packings decorrelate as well as the correlations between the density maxima as the space dimension increases is also investigated. We find that the extreme lattices decorrelate with increasing dimension, the least symmetric lattices decorrelate faster. The extreme lattices in a fixed dimension of space $d$ ($d\geq 8$) are dominated by typical lattices that have similar packing properties, like packing densities and kissing numbers, while the best and the worst packers are in the long tails of the distribution of the extreme lattices.

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arXiv2013

An Efficient Linear Programming Algorithm to Generate the Densest Lattice Sphere Packings

Finding the densest sphere packing in $d$-dimensional Euclidean space $\mathbb{R}^d$ is an outstanding fundamental problem with relevance in many fields, including the ground states of molecular systems, colloidal crystal structures, coding theory, discrete geometry, number theory, and biological systems. Numerically generating the densest sphere packings becomes very challenging in high dimensions due to an exponentially increasing number of possible sphere contacts and sphere configurations, even for the restricted problem of finding the densest lattice sphere packings. In this paper, we apply the Torquato-Jiao packing algorithm, which is a method based on solving a sequence of linear programs, to robustly reproduce the densest known lattice sphere packings for dimensions 2 through 19. We show that the TJ algorithm is appreciably more efficient at solving these problems than previously published methods. Indeed, in some dimensions, the former procedure can be as much as three orders of magnitude faster at finding the optimal solutions than earlier ones. We also study the suboptimal local density-maxima solutions (inherent structures or "extreme" lattices) to gain insight about the nature of the topography of the "density" landscape.

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arXiv2013

Effect of Dimensionality on the Percolation Thresholds of Various $d$-Dimensional Lattices

We show analytically that the $[0,1]$, $[1,1]$ and $[2,1]$ Pad{é} approximants of the mean cluster number $S(p)$ for site and bond percolation on general $d$-dimensional lattices are upper bounds on this quantity in any Euclidean dimension $d$, where $p$ is the occupation probability. These results lead to certain lower bounds on the percolation threshold $p_c$ that become progressively tighter as $d$ increases and asymptotically exact as $d$ becomes large. These lower-bound estimates depend on the structure of the $d$-dimensional lattice and whether site or bond percolation is being considered. We obtain explicit bounds on $p_c$ for both site and bond percolation on five different lattices: $d$-dimensional generalizations of the simple-cubic, body-centered-cubic and face-centered-cubic Bravais lattices as well as the $d$-dimensional generalizations of the diamond and kagom{é} (or pyrochlore) non-Bravais lattices. These analytical estimates are used to assess available simulation results across dimensions (up through $d=13$ in some cases). It is noteworthy that the tightest lower bound provides reasonable estimates of $p_c$ in relatively low dimensions and becomes increasingly accurate as $d$ grows. We also derive high-dimensional asymptotic expansions for $p_c$ for the ten percolation problems and compare them to the Bethe-lattice approximation. Finally, we remark on the radius of convergence of the series expansion of $S$ in powers of $p$ as the dimension grows.

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arXiv2013

Effect of Dimensionality on the Percolation Threshold of Overlapping Nonspherical Hyperparticles

A set of lower bounds on the continuum percolation threshold $η_c$ of overlapping convex hyperparticles of general nonspherical (anisotropic) shape with a specified orientational probability distribution in $d$-dimensional Euclidean space have been derived [S. Torquato, J. Chem. Phys. {\bf 136}, 054106 (2012)]. The simplest of these lower bounds is given by $η_c \ge v/v_{ex}$, where $v_{ex}$ is the $d$-dimensional exclusion volume of a hyperparticle and $v$ is its $d$-dimensional volume. In order to study the effect of dimensionality on the threshold $η_c$ of overlapping nonspherical convex hyperparticles with random orientations here, we obtain a scaling relation for $η_c$ that is based on this lower bound and a conjecture that hyperspheres provide the highest threshold among all convex hyperparticle shapes for any $d$. This scaling relation exploits the principle that low-dimensional continuum percolation behavior encodes high-dimensional information. We derive a formula for the exclusion volume $v_{ex}$ of a hyperparticle in terms of its $d$-dimensional volume $v$, surface area $s$ and {\it radius of mean curvature} ${\bar R}$ (or, equivalently, {\it mean width}). These basic geometrical properties are computed for a wide variety of nonspherical hyperparticle shapes with random orientations across all dimensions, including, among other shapes, various polygons for $d=2$, Platonic solids, spherocylinders, parallepipeds and zero-volume plates for $d=3$ and their appropriate generalizations for $d \ge 4$. We then compute the lower bound and scaling relation for $η_c$ for this comprehensive set of continuum percolation models across dimensions. We show that the scaling relation provides accurate {\it upper-bound} estimates of the threshold $η_c$ across dimensions and becomes increasingly accurate as the space $d$ increases.

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arXiv2012

Designed Diamond Ground State via Optimized Isotropic Monotonic Pair Potentials

We apply inverse statistical-mechanical methods to find a simple family of optimized isotropic, monotonic pair potentials, under certain constraints, whose ground states for a wide range of pressures is the diamond crystal. These constraints include desirable phonon spectra and the widest possible pressure range for stability. We also ascertain the ground-state phase diagram for a specific optimized potential to show that other crystal structures arise for other pressures. Cooling disordered configurations interacting with our optimized potential to absolute zero frequently leads to the desired diamond crystal ground state, revealing that the capture basin for the global energy minimum is large and broad relative to the local energy minima basins.

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arXiv2012

Exotic Ground States of Directional Pair Potentials via Collective-Density Variables

Collective-density variables have proved to be a useful tool in the prediction and manipulation of how spatial patterns form in the classical many-body problem. Previous work has employed properties of collective-density variables along with a robust numerical optimization technique to find the classical ground states of many-particle systems subject to radial pair potentials in one, two and three dimensions. That work led to the identification of ordered and disordered classical ground states. In this paper, we extend these collective-coordinate studies by investigating the ground states of directional pair potentials in two dimensions. Our study focuses on directional potentials whose Fourier representations are non-zero on compact sets that are symmetric with respect to the origin and zero everywhere else. We choose to focus on one representative set which has exotic ground-state properties: two circles whose centers are separated by some fixed distance. We obtain ground states for this "two-circle" potential that display large void regions in the disordered regime. As more degrees of freedom are constrained the ground states exhibit a collapse of dimensionality characterized by the emergence of filamentary structures and linear chains. This collapse of dimensionality has not been observed before in related studies.

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arXiv2012

Nonequilibrium static growing length scales in supercooled liquids on approaching the glass transition

The small wavenumber $k$ behavior of the structure factor $S(k)$ of overcompressed amorphous hard-sphere configurations was previously studied for a wide range of densities up to the maximally random jammed state, which can be viewed as a prototypical glassy state [A. Hopkins, F. H. Stillinger and S. Torquato, Phys. Rev. E, 86, 021505 (2012)]. It was found that a precursor to the glassy jammed state was evident long before the jamming density was reached as measured by a growing nonequilibrium length scale extracted from the volume integral of the direct correlation function $c(r)$, which becomes long-ranged as the critical jammed state is reached. The present study extends that work by investigating via computer simulations two different atomic models: the single-component Z2 Dzugutov potential in three dimensions and the binary-mixture Kob-Andersen potential in two dimensions. Consistent with the aforementioned hard-sphere study, we demonstrate that for both models a signature of the glass transition is apparent well before the transition temperature is reached as measured by the length scale determined from from the volume integral of the direct correlation function in the single-component case and a generalized direct correlation function in the binary-mixture case. The latter quantity is obtained from a generalized Orstein-Zernike integral equation for a certain decoration of the atomic point configuration. We also show that these growing length scales, which are a consequence of the long-range nature of the direct correlation functions, are intrinsically nonequilibrium in nature as determined by an index $X$ that is a measure of deviation from thermal equilibrium. It is also demonstrated that this nonequilibrium index, which increases upon supercooling, is correlated with a characteristic relaxation time scale.

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arXiv2012

Effect of Dimensionality on the Continuum Percolation of Overlapping Hyperspheres and Hypercubes: II. Simulation Results and Analyses

In the first paper of this series [S. Torquato, J. Chem. Phys. {\bf 136}, 054106 (2012)], analytical results concerning the continuum percolation of overlapping hyperparticles in $d$-dimensional Euclidean space $\mathbb{R}^d$ were obtained, including lower bounds on the percolation threshold. In the present investigation, we provide additional analytical results for certain cluster statistics, such as the concentration of $k$-mers and related quantities, and obtain an upper bound on the percolation threshold $η_c$. We utilize the tightest lower bound obtained in the first paper to formulate an efficient simulation method, called the {\it rescaled-particle} algorithm, to estimate continuum percolation properties across many space dimensions with heretofore unattained accuracy. This simulation procedure is applied to compute the threshold $η_c$ and associated mean number of overlaps per particle ${\cal N}_c$ for both overlapping hyperspheres and oriented hypercubes for $ 3 \le d \le 11$. These simulations results are compared to corresponding upper and lower bounds on these percolation properties. We find that the bounds converge to one another as the space dimension increases, but the lower bound provides an excellent estimate of $η_c$ and ${\cal N}_c$, even for relatively low dimensions. We confirm a prediction of the first paper in this series that low-dimensional percolation properties encode high-dimensional information. We also show that the concentration of monomers dominate over concentration values for higher-order clusters (dimers, trimers, etc.) as the space dimension becomes large. Finally, we provide accurate analytical estimates of the pair connectedness function and blocking function at their contact values for any $d$ as a function of density.

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arXiv2012

Organizing Principles for Dense Packings of Nonspherical Hard Particles: Not All Shapes Are Created Equal

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. We apply and test all of these organizing principles to the most comprehensive set of both convex and concave particle shapes 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.

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