arXiv subjects

Salvatore Torquato

Publications and source records attributed to Salvatore Torquato.

179 records · Page 10Linked to original sources

arXiv2007

Effective Dielectric Tensor for Electromagnetic Wave Propagation in Random Media

We derive exact strong-contrast expansions for the effective dielectric tensor $\epeff$ of electromagnetic waves propagating in a two-phase composite random medium with isotropic components explicitly in terms of certain integrals over the $n$-point correlation functions of the medium. Our focus is the long-wavelength regime, i.e., when the wavelength is much larger than the scale of inhomogeneities in the medium. Lower-order truncations of these expansions lead to approximations for the effective dielectric constant that depend upon whether the medium is below or above the percolation threshold. In particular, we apply two- and three-point approximations for $\epeff$ to a variety of different three-dimensional model microstructures, including dispersions of hard spheres, hard oriented spheroids and fully penetrable spheres as well as Debye random media, the random checkerboard, and power-law-correlated materials. We demonstrate the importance of employing $n$-point correlation functions of order higher than two for high dielectric-phase-contrast ratio. We show that disorder in the microstructure results in an imaginary component of the effective dielectric tensor that is directly related to the {\it coarseness} of the composite, i.e., local volume-fraction fluctuations for infinitely large windows. The source of this imaginary component is the attenuation of the coherent homogenized wave due to scattering. We also remark on whether there is such attenuation in the case of a two-phase medium with a quasiperiodic structure.

cond-mat.mtrl-sci↗

arXiv2006

Self-assembly of the simple cubic lattice with an isotropic potential

Conventional wisdom presumes that low-coordinated crystal ground states require directional interactions. Using our recently introduced optimization procedure to achieve self-assembly of targeted structures (Phys. Rev. Lett. 95, 228301 (2005), Phys. Rev. E 73, 011406 (2006)), we present an isotropic pair potential $V(r)$ for a three-dimensional many-particle system whose classical ground state is the low-coordinated simple cubic (SC) lattice. This result is part of an ongoing pursuit by the authors to develop analytical and computational tools to solve statistical-mechanical inverse problems for the purpose of achieving targeted self-assembly. The purpose of these methods is to design interparticle interactions that cause self-assembly of technologically important target structures for applications in photonics, catalysis, separation, sensors and electronics. We also show that standard approximate integral-equation theories of the liquid state that utilize pair correlation function information cannot be used in the reverse mode to predict the correct simple cubic potential. We report in passing optimized isotropic potentials that yield the body-centered cubic and simple hexagonal lattices, which provide other examples of non-close-packed structures that can be assembled using isotropic pair interactions.

cond-mat.soft↗

arXiv2006

Necessary Conditions on Realizable Two-Point Correlation Functions of Random Media

A fascinating inverse problem that has been receiving considerable attention is the construction of realizations of random two-phase heterogeneous media with a target two-point correlation function. However, not every hypothetical two-point correlation function corresponds to a realizable two-phase medium. Here we collect all of the known necessary conditions on the two-point correlation functions scattered throughout a diverse literature and derive a new but simple positivity condition. We apply the necessary conditions to test the realizability of certain classes of proposed correlation functions.

cond-mat.dis-nn↗

arXiv2006

Collective Coordinate Control of Density Distributions

Real collective density variables $C(\boldsymbol{k})$ [c.f. Eq.\ref{Equation3})] in many-particle systems arise from non-linear transformations of particle positions, and determine the structure factor $S(\boldsymbol{k})$, where $\bf k$ denotes the wave vector. Our objective is to prescribe $C({\boldsymbol k})$ and then to find many-particle configurations that correspond to such a target $C({\bf k})$ using a numerical optimization technique. Numerical results reported here extend earlier one- and two-dimensional studies to include three dimensions. In addition, they demonstrate the capacity to control $S(\boldsymbol{k})$ in the neighborhood of $|\boldsymbol{k}| =$ 0. The optimization method employed generates multi-particle configurations for which $S(\boldsymbol{k}) \propto |\boldsymbol{k}|^α$, $|\boldsymbol{k}| \leq K$, and $α=$ 1, 2, 4, 6, 8, and 10. The case $α=$ 1 is relevant for the Harrison-Zeldovich model of the early universe, for superfluid $^{4}{He}$, and for jammed amorphous sphere packings. The analysis also provides specific examples of interaction potentials whose classical ground state are configurationally degenerate and disordered.

cond-mat.stat-mech↗

arXiv2006

A Variational Level Set Approach for Surface Area Minimization of Triply Periodic Surfaces

In this paper, we study triply periodic surfaces with minimal surface area under a constraint in the volume fraction of the regions (phases) that the surface separates. Using a variational level set method formulation, we present a theoretical characterization of and a numerical algorithm for computing these surfaces. We use our theoretical and computational formulation to study the optimality of the Schwartz P, Schwartz D, and Schoen G surfaces when the volume fractions of the two phases are equal and explore the properties of optimal structures when the volume fractions of the two phases not equal. Due to the computational cost of the fully, three-dimensional shape optimization problem, we implement our numerical simulations using a parallel level set method software package.

math.OC↗

arXiv2006

Designed Interaction Potentials via Inverse Methods for Self-Assembly

We formulate statistical-mechanical inverse methods in order to determine optimized interparticle interactions that spontaneously produce target many-particle configurations. Motivated by advances that give experimentalists greater and greater control over colloidal interaction potentials, we propose and discuss two computational algorithms that search for optimal potentials for self-assembly of a given target configuration. The first optimizes the potential near the ground state and the second near the melting point. We begin by applying these techniques to assembling open structures in two dimensions (square and honeycomb lattices) using only circularly symmetric pair interaction potentials ; we demonstrate that the algorithms do indeed cause self-assembly of the target lattice. Our approach is distinguished from previous work in that we consider (i) lattice sums, (ii) mechanical stability (phonon spectra), and (iii) annealed Monte Carlo simulations. We also devise circularly symmetric potentials that yield chain-like structures as well as systems of clusters.

cond-mat.soft↗

arXiv2005

Optimized Interactions for Targeted Self-Assembly: Application to Honeycomb Lattice

We devise an inverse statistical-mechanical methodology to find optimized interaction potentials that lead spontaneously to a target many-particle configuration. Target structures can possess varying degrees of disorder, thus extending the traditional idea of self-assembly to incorporate both amorphous and crystalline structures as well as quasicrystals. For illustration purposes, our computational technique is applied to yield an optimized isotropic (non-directional) pair potential that spontaneously yields the three-coordinated honeycomb lattice as the ground state structure in two dimensions. This target choice is motivated by its three-dimensional analog, the diamond lattice, which is known to possess desirable photonic bandgap properties.

cond-mat.soft↗

arXiv2004

Neighbor List Collision-Driven Molecular Dynamics Simulation for Nonspherical Particles. I. Algorithmic Details II. Applications to Ellipses and Ellipsoids

In the first part of a series of two papers, we present in considerable detail a collision-driven molecular dynamics algorithm for a system of nonspherical particles, within a parallelepiped simulation domain, under both periodic or hard-wall boundary conditions. The algorithm extends previous event-driven molecular dynamics algorithms for spheres. We present a novel partial-update near-neighbor list (NNL) algorithm that is superior to previous algorithms at high densities, without compromising the correctness of the algorithm. This efficiency of the algorithm is further increased for systems of very aspherical particles by using bounding sphere complexes (BSC). In the second part of this series of papers we apply the algorithm presented in the first part of this series of papers to systems of hard ellipses and ellipsoids. The theoretical machinery needed to treat such particles, including the overlap potentials, is developed in full detail. We describe an algorithm for predicting the time of collision for two moving ellipses or ellipsoids. We present performance results for our implementation of the algorithm. The practical utility of the algorithm is demonstrated by presenting several interesting physical applications, including the generation of jammed packings inside spherical containers, the study of contact force chains in jammed packings, and melting the densest-known equilibrium crystals of prolate spheroids.

physics.comp-ph↗

arXiv2004

Voronoi and Voids Statistics for Super-homogeneous Point Processes

We study the Voronoi and void statistics of super-homogeneous (or hyperuniform) point patterns in which the infinite-wavelength density fluctuations vanish. Super-homogeneous or hyperuniform point patterns arise in one-component plasmas, primordial density fluctuations in the Universe, and in jammed hard-particle packings. We specifically analyze a certain one-dimensional model by studying size fluctuations and correlations of the associated Voronoi cells. We derive exact results for the complete joint statistics of the size of two Voronoi cells. We also provide a sum rule that the correlation matrix for the Voronoi cells must obey in any space dimension. In contrast to the conventional picture of super-homogeneous systems, we show that infinitely large Voronoi cells or voids can exist in super-homogeneous point processes in any dimension. We also present two heuristic conditions to identify and classify any super-homogeneous point process in terms of the asymptotic behavior of the void size distribution.

cond-mat.stat-mech↗

arXiv2004

Superdense Crystal Packings of Ellipsoids

Particle packing problems have fascinated people since the dawn of civilization, and continue to intrigue mathematicians and scientists. Resurgent interest has been spurred by the recent proof of Kepler's conjecture: the face-centered cubic lattice provides the densest packing of equal spheres with a packing fraction $ϕ\approx0.7405$ \cite{Kepler_Hales}. Here we report on the densest known packings of congruent ellipsoids. The family of new packings are crystal (periodic) arrangements of nearly spherically-shaped ellipsoids, and always surpass the densest lattice packing. A remarkable maximum density of $ϕ\approx0.7707$ is achieved for both prolate and oblate ellipsoids with aspect ratios of $\sqrt{3}$ and $1/\sqrt{3}$, respectively, and each ellipsoid has 14 touching neighbors. Present results do not exclude the possibility that even denser crystal packings of ellipsoids could be found, and that a corresponding Kepler-like conjecture could be formulated for ellipsoids.

cond-mat.stat-mech↗

arXiv2003

Local Density Fluctuations, Hyperuniformity, and Order Metrics

We study the variance in the number of points contained within a window $Ω$ of arbitrary size, and to further illuminate our understanding of {\it hyperuniform} systems, i.e., point patterns that do not possess long-wavelength fluctuations. For large windows, hyperuniform systems are characterized by a local variance that grows only as the surface area (rather than the volume) of the window. We show that a homogeneous point pattern in a hyperuniform state is at a ``critical-point'' of a type with appropriate scaling laws and critical exponents, but one in which the {\it direct correlation function} (rather than the pair correlation function) is long-ranged.variance.We prove that the simple periodic linear array yields the global minimum value of the average variance among all infinite one-dimensional hyperuniform patterns. We also evaluate the variance for common infinite periodic lattices as well as certain nonperiodic point patterns in one, two, and three dimensions for spherical windows.Our results suggest that the local variance may serve as a useful order metric for general point patterns. Contrary to the conjecture that the lattices associated with the densest packing of congruent spheres have the smallest variance regardless of the space dimension, we show that for $d=3$, the body-centered cubic lattice has a smaller variance than the face-centered cubic lattice. Finally, for certain hyperuniform disordered point patterns, we evaluate the direct correlation function, structure factor, and associated critical exponents exactly.

cond-mat.stat-mech↗

arXiv2002

A Linear Programming Algorithm to Test for Jamming in Hard-Sphere Packings

Jamming in hard-particle packings has been the subject of considerable interest in recent years. In a paper by Torquato and Stillinger [J. Phys. Chem. B, 105 (2001)], a classification scheme of jammed packings into hierarchical categories of locally, collectively and strictly jammed configurations has been proposed. They suggest that these jamming categories can be tested using numerical algorithms that analyze an equivalent contact network of the packing under applied displacements, but leave the design of such algorithms as a future task. In this work we present a rigorous and efficient algorithm to assess whether a hard-sphere packing is jammed according to the afformentioned categories. The algorithm is based on linear programming and is applicable to regular as well as random packings of finite size with hard-wall and periodic boundary conditions. If the packing is not jammed, the algorithm yields representative multi-particle unjamming motions. We have implemented this algorithm and applied it to ordered lattices as well as random packings of disks and spheres under periodic boundary conditions. Some representative results for ordered and disordered packings are given, but more applications are anticipated for the future. One important and interesting result is that the random packings that we tested were strictly jammed in three dimensions, but not in two dimensions. Numerous interactive visualization models are provided on the authors' webpage.

cond-mat.mtrl-sci↗

arXiv2002

Quantification of Order in the Lennard-Jones System

We conduct a numerical investigation of structural order in the shifted-force Lennard-Jones system by calculating metrics of translational and bond-orientational order along various paths in the phase diagram covering equilibrium solid, liquid, and vapor states. A series of non-equilibrium configurations generated through isochoric quenches, isothermal compressions, and energy minimizations are also considered. Simulation results are analyzed using an ordering map representation [Torquato et al., Phys. Rev. Lett. 84, 2064 (2000); Truskett et al., Phys. Rev. E 62, 993 (2000)] that assigns to both equilibrium and non-equilibrium states coordinates in an order metric plane. Our results show that bond-orientational order and translational order are not independent for simple spherically symmetric systems at equilibrium. We also demonstrate quantitatively that the Lennard-Jones and hard sphere systems sample the same configuration space at supercritical densities. Finally, we relate the structural order found in fast-quenched and minimum-energy configurations (inherent structures).

cond-mat.stat-mech↗

arXiv2002

Cooperative Origin of Low-Density Domains in Liquid Water

We study the size of clusters formed by water molecules possessing large enough tetrahedrality with respect to their nearest neighbors. Using Monte Carlo simulation of the SPC/E model of water, together with a geometric analysis based on Voronoi tessellation, we find that regions of lower density than the bulk are formed by accretion of molecules into clusters exceeding a minimum size. Clusters are predominantly linear objects and become less compact as they grow until they reach a size beyond which further accretion is not accompanied by a density decrease. The results suggest that the formation of "ice-like" regions in liquid water is cooperative.

cond-mat.stat-mech↗

arXiv2001

Globally and Locally Minimal Weight Spanning Tree Networks

The competition between local and global driving forces is significant in a wide variety of naturally occurring branched networks. We have investigated the impact of a global minimization criterion versus a local one on the structure of spanning trees. To do so, we consider two spanning tree structures - the generalized minimal spanning tree (GMST) defined by Dror et al. [1] and an analogous structure based on the invasion percolation network, which we term the generalized invasive spanning tree or GIST. In general, these two structures represent extremes of global and local optimality, respectively. Structural characteristics are compared between the GMST and GIST for a fixed lattice. In addition, we demonstrate a method for creating a series of structures which enable one to span the range between these two extremes. Two structural characterizations, the occupied edge density (i.e., the fraction of edges in the graph that are included in the tree) and the tortuosity of the arcs in the trees, are shown to correlate well with the degree to which an intermediate structure resembles the GMST or GIST. Both characterizations are straightforward to determine from an image and are potentially useful tools in the analysis of the formation of network structures.

cond-mat.dis-nn↗

arXiv2000

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.

cond-mat.stat-mech↗

arXiv1999

Triangle Distribution and Equation of State for Classical Rigid Disks

The triangle distribution function f^(3) for three mutual nearest neighbors in the plane describes basic aspects of short-range order and statistical thermodynamics in two-dimensional many-particle systems. This paper examines prospects for constructing a self-consistent calculation for the rigid-disk system f^(3). We present several identities obeyed by f^(3). A rudimentary closure suggested by scaled-particle theory is introduced. In conjunction with three of the basic identities, this closure leads to a unique f^(3) over the entire density range. The pressure equation of state exhibits qualitatively correct behavior in both the low density and the close-packed limits, but no intervening phase transition appears. We discuss extensions to improved disk closures, and to the three-dimensional rigid sphere system.

cond-mat.stat-mech↗