Biomedical subjects

Salvatore Torquato

Publications and source records attributed to Salvatore Torquato.

At least 19 recordsLinked to original sources

PubMed2006

Packing hyperspheres in high-dimensional Euclidean spaces.

We present a study of disordered jammed hard-sphere packings in four-, five-, and six-dimensional Euclidean spaces. Using a collision-driven packing generation algorithm, we obtain the first estimates for the packing fractions of the maximally random jammed (MRJ) states for space dimensions d=4, 5, and 6 to be phi(MRJ) approximately 0.46, 0.31, and 0.20, respectively. To a good approximation, the MRJ density obeys the scaling form phi(MRJ)=c1/2(d)+(c2d)/2d, where c1=-2.72 and c2=2.56, which appears to be consistent with the high-dimensional asymptotic limit, albeit with different coefficients. Calculations of the pair correlation function g2(r) and structure factor S(k) for these states show that short-range ordering appreciably decreases with increasing dimension, consistent with a recently proposed "decorrelation principle," which, among other things, states that unconstrained correlations diminish as the dimension increases and vanish entirely in the limit d-->infinity. As in three dimensions (where phi(MRJ) approximately 0.64), the packings show no signs of crystallization, are isostatic, and have a power-law divergence in g2(r) at contact with power-law exponent approximately 0.4. Across dimensions, the cumulative number of neighbors equals the kissing number of the conjectured densest packing close to where g2(r) has its first minimum. Additionally, we obtain estimates for the freezing and melting packing fractions for the equilibrium hard-sphere fluid-solid transition, phi(F) approximately 0.32 and phi(M) approximately 0.39, respectively, for d=4, and phi(F) approximately 0.20 and phi(M) approximately 0.25, respectively, for d=5. Although our results indicate the stable phase at high density is a crystalline solid, nucleation appears to be strongly suppressed with increasing dimension.

Journal Article↗

PubMed2006

Collective coordinate control of density distributions.

Real collective density variables C(k) [cf. Eq. 1 3 ] in many-particle systems arise from nonlinear transformations of particle positions, and determine the structure factor S(k) , where k denotes the wave vector. Our objective is to prescribe C(k) and then to find many-particle configurations that correspond to such a target C(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(k) in the neighborhood of |k|=0. The optimization method employed generates multiparticle configurations for which S(k) proportional, |k|alpha, |k|<or=K, and alpha=1, 2, 4, 6, 8, and 10. The case alpha=1 is relevant for the Harrison-Zeldovich model of the early universe, for superfluid 4He, and for jammed amorphous sphere packings. The analysis also provides specific examples of interaction potentials whose classical ground states are configurationally degenerate and disordered.

Journal Article↗

PubMed2006

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 [M. C. Rechtsman, 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 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.

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PubMed2006

Modeling the effects of vasculature evolution on early brain tumor growth.

Mathematical modeling of both tumor growth and angiogenesis have been active areas of research for the past several decades. Such models can be classified into one of two categories: those that analyze the remodeling of the vasculature while ignoring changes in the tumor mass, and those that predict tumor expansion in the presence of a non-evolving vasculature. However, it is well accepted that vasculature remodeling and tumor growth strongly depend on one another. For this reason, we have developed a two-dimensional hybrid cellular automaton model of early brain tumor growth that couples the remodeling of the microvasculature with the evolution of the tumor mass. A system of reaction-diffusion equations has been developed to track the concentration of vascular endothelial growth factor (VEGF), Ang-1, Ang-2, their receptors and their complexes in space and time. The properties of the vasculature and hence of each cell are determined by the relative concentrations of these key angiogenic factors. The model exhibits an angiogenic switch consistent with experimental observations on the upregulation of angiogenesis. Particularly, we show that if the pathways that produce and respond to VEGF and the angiopoietins are properly functioning, angiogenesis is initiated and a tumor can grow to a macroscopic size. However, if the VEGF pathway is inhibited, angiogenesis does not occur and tumor growth is thwarted beyond 1-2mm in size. Furthermore, we show that tumor expansion can occur in well-vascularized environments even when angiogenesis is inhibited, suggesting that anti-angiogenic therapies may not be sufficient to eliminate a population of actively dividing malignant cells.

Algorithms↗

PubMed2006

Global phase diagram for the honeycomb potential.

We calculate the global phase diagram using classical statistical mechanics for an isotropic pair potential that has been previously [Rechtsman et al., Phys. Rev. Lett. 95, 228301 (2005)] shown to produce the low-coordinated two-dimensional honeycomb crystal as the ground-state structure. Low-coordinated crystals are of practical interest because they have desirable photonic band-gap properties. The phase diagram is obtained from Helmholtz free energies calculated using thermodynamic integration and Monte Carlo simulations. Our results show that the honeycomb crystal remains stable in the global phase diagram even after temperature effects are taken fully into account. Other stable phases in the phase diagram are high and low density triangular phases and a fluid phase. We find no evidence of gas-liquid or liquid-liquid phase coexistence.

Journal Article↗

PubMed2006

Do binary hard disks exhibit an ideal glass transition?

We demonstrate that there is no ideal glass transition in a binary hard-disk mixture by explicitly constructing an exponential number of jammed packings with densities spanning the spectrum from the accepted amorphous glassy state to the phase-separated crystal. Thus the configurational entropy cannot be zero for an ideal amorphous glass, presumed distinct from the crystal in numerous theoretical and numerical estimates in the literature. This objection parallels our previous critique of the idea that there is a most-dense random (close) packing for hard spheres [Torquato, Phys. Rev. Lett. 84, 2064 (2000)10.1103/PhysRevLett.84.2064].

Journal Article↗

PubMed2006

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 chainlike structures as well as systems of clusters.

Journal Article↗

PubMed2005

Optimized interactions for targeted self-assembly: application to a 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 (nondirectional) 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 band gap properties.

Biomechanical Phenomena↗

PubMed2005

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).

Journal Article↗

PubMed2005

Experiments on random packings of ellipsoids.

Recent simulations indicate that ellipsoids can pack randomly more densely than spheres and, remarkably, for axes ratios near 1.25:1:0.8 can approach the densest crystal packing (fcc) of spheres, with a packing fraction of 74%. We demonstrate that such dense packings are realizable. We introduce a novel way of determining packing density for a finite sample that minimizes surface effects. We have fabricated ellipsoids and show that, in a sphere, the radial packing fraction phi(r) can be obtained from V(h), the volume of added fluid to fill the sphere to height h. We also obtain phi(r) from a magnetic resonance imaging scan. The measurements of the overall density phi(avr), phi(r) and the core density phi(0) = 0.74 +/- 0.005 agree with simulations.

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PubMed2005

Pair correlation function characteristics of nearly jammed disordered and ordered hard-sphere packings.

We study the approach to jamming in hard-sphere packings and, in particular, the pair correlation function g(2) (r) around contact, both theoretically and computationally. Our computational data unambiguously separate the narrowing delta -function contribution to g(2) due to emerging interparticle contacts from the background contribution due to near contacts. The data also show with unprecedented accuracy that disordered hard-sphere packings are strictly isostatic: i.e., the number of exact contacts in the jamming limit is exactly equal to the number of degrees of freedom, once rattlers are removed. For such isostatic packings, we derive a theoretical connection between the probability distribution of interparticle forces P(f) (f) , which we measure computationally, and the contact contribution to g(2) . We verify this relation for computationally generated isostatic packings that are representative of the maximally random jammed state. We clearly observe a maximum in P(f) and a nonzero probability of zero force, shedding light on long-standing questions in the granular-media literature. We computationally observe an unusual power-law divergence in the near-contact contribution to g(2) , persistent even in the jamming limit, with exponent -0.4 clearly distinguishable from previously proposed inverse-square-root divergence. Additionally, we present high-quality numerical data on the two discontinuities in the split-second peak of g(2) and use a shared-neighbor analysis of the graph representing the contact network to study the local particle clusters responsible for the peculiar features. Finally, we present the computational data on the contact contribution to g(2) for vacancy-diluted fcc crystal packings and also investigate partially crystallized packings along the transition from maximally disordered to fully ordered packings. We find that the contact network remains isostatic even when ordering is present. Unlike previous studies, we find that ordering has a significant impact on the shape of P(f) for small forces.

Journal Article↗

PubMed2004

Constraints on collective density variables: two dimensions.

Collective density variables rho (k) have proved to be useful tools in the study of many-body problems in a variety of fields that are concerned with structural and kinematic phenomena. In spite of their broad applicability, mathematical understanding of collective density variables remains an underexplored subject. In this paper, we examine features associated with collective density variables in two dimensions using numerical exploration techniques to generate particle patterns in the classical ground state. Particle pair interactions are governed by a continuous, bounded potential. Our approach involves constraining related collective parameters C (k) , with wave vector k magnitudes at or below a chosen cutoff, to their absolute minimum values. Density fluctuations for those k 's thus are suppressed. The resulting investigation distinguishes three structural regimes as the number of constrained wave vectors is increased-disordered, wavy crystalline, and crystalline regimes-each with characteristic distinguishing features. It should be noted that our choice of pair potential can lead to pair correlation functions that exhibit an effective hard core and thus signal the formation of a hard-disk-like equilibrium fluid. In addition, our method creates particle patterns that are hyperuniform, thus supporting the notion that structural glasses can be hyperuniform as the temperature T-->0 .

Journal Article↗

PubMed2004

Voronoi and void statistics for superhomogeneous point processes.

We study the Voronoi and void statistics of superhomogeneous (or hyperuniform) point patterns in which the infinite-wavelength density fluctuations vanish. Superhomogeneous or hyperuniform point patterns arise in one-component plasmas, primordial density fluctuations in the Universe, and 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 superhomogeneous systems, we show that infinitely large Voronoi cells or voids can exist in superhomogeneous point processes in any dimension. We also present two heuristic conditions to identify and classify any superhomogeneous point process in terms of the asymptotic behavior of the void size distribution.

Journal Article↗

PubMed2004

Comment on "Jamming at zero temperature and zero applied stress: The epitome of disorder".

Phys. Rev. E. 68, 011306 (2003)] claim that a special point J of a "jamming phase diagram" (in density, temperature, stress space) is related to random close packing of hard spheres and that it represents, for their suggested definitions of jammed and random, the recently introduced maximally random jammed state. We point out several difficulties with their definitions and question some of their claims. Furthermore, we discuss the connections between their algorithm and other hard-sphere packing algorithms in the literature.

Comment↗

PubMed2004

Unusually dense crystal packings of ellipsoids.

In this Letter, we report on the densest-known packings of congruent ellipsoids. The family of new packings consists of crystal arrangements of spheroids with a wide range of aspect ratios, and with density phi always surpassing that of the densest Bravais lattice packing phi approximately equal to 0.7405. A remarkable maximum density of phi approximately equal to 0.7707 is achieved for maximal aspect ratios larger than sqrt[3], when each ellipsoid has 14 touching neighbors. Our results are directly relevant to understanding the equilibrium behavior of systems of hard ellipsoids, and, in particular, the solid and glassy phases.

Journal Article↗

PubMed2004

Improving the density of jammed disordered packings using ellipsoids.

Packing problems, such as how densely objects can fill a volume, are among the most ancient and persistent problems in mathematics and science. For equal spheres, it has only recently been proved that the face-centered cubic lattice has the highest possible packing fraction phi=pi/18 approximately 0.74. It is also well known that certain random (amorphous) jammed packings have phi approximately 0.64. Here, we show experimentally and with a new simulation algorithm that ellipsoids can randomly pack more densely-up to phi= 0.68 to 0.71 for spheroids with an aspect ratio close to that of M&M's Candies-and even approach phi approximately 0.74 for ellipsoids with other aspect ratios. We suggest that the higher density is directly related to the higher number of degrees of freedom per particle and thus the larger number of particle contacts required to mechanically stabilize the packing. We measured the number of contacts per particle Z approximately 10 for our spheroids, as compared to Z approximately 6 for spheres. Our results have implications for a broad range of scientific disciplines, including the properties of granular media and ceramics, glass formation, and discrete geometry.

Journal Article↗

PubMed2003

Local density fluctuations, hyperuniformity, and order metrics.

Questions concerning the properties and quantification of density fluctuations in point patterns continue to provide many theoretical challenges. The purpose of this paper is to characterize certain fundamental aspects of local density fluctuations associated with general point patterns in any space dimension d. Our specific objectives are to study the variance in the number of points contained within a regularly shaped window Omega of arbitrary size, and to further illuminate our understanding of hyperuniform systems, i.e., point patterns that do not possess infinite-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 derive two formulations for the number variance: (i) an ensemble-average formulation, which is valid for statistically homogeneous systems, and (ii) a volume-average formulation, applicable to a single realization of a general point pattern in the large-system limit. The ensemble-average formulation (which includes both real-space and Fourier representations) enables us to 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 direct correlation function (rather than the pair correlation function) is long ranged. We also prove that the non-negativity of the local number variance does not add a new realizability condition on the pair correlation. The volume-average formulation is superior for certain computational purposes, including optimization studies in which it is desired to find the particular point pattern with an extremal or targeted value of the 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, enabling us to rank-order the spatial patterns. 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.

Journal Article↗

PubMed2002

Cooperative origin of low-density domains in liquid water.

We study clusters formed by water molecules possessing large enough tetrahedrality with respect to their nearest neighbors. Using Monte Carlo simulation of SPC/E water and 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 "icelike" regions in liquid water is cooperative.

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