arXiv subjects

Salvatore Torquato

Publications and source records attributed to Salvatore Torquato.

At least 163 records · Page 9Linked to original sources

arXiv2010

Novel Ground-State Crystals with Controlled Vacancy Concentrations: From Kagomé to Honeycomb to Stripes

We introduce a one-parameter family, $0 \leq H \leq 1$, of pair potential functions with a single relative energy minimum that stabilize a range of vacancy-riddled crystals as ground states. The "quintic potential" is a short-ranged, nonnegative pair potential with a single local minimum of height $H$ at unit distance and vanishes cubically at a distance of $\rt$. We have developed this potential to produce ground states with the symmetry of the triangular lattice while favoring the presence of vacancies. After an exhaustive search using various optimization and simulation methods, we believe that we have determined the ground states for all pressures, densities, and $0 \leq H \leq 1$. For specific areas below $3\rt/2$, the ground states of the "quintic potential" include high-density and low-density triangular lattices, kagomé and honeycomb crystals, and stripes. We find that these ground states are mechanically stable but are difficult to self-assemble in computer simulations without defects. For specific areas above $3\rt/2$, these systems have a ground-state phase diagram that corresponds to hard disks with radius $\rt$. For the special case of H=0, a broad range of ground states is available. Analysis of this case suggests that among many ground states, a high-density triangular lattice, low-density triangular lattice, and striped phases have the highest entropy for certain densities. The simplicity of this potential makes it an attractive candidate for experimental realization with application to the development of novel colloidal crystals or photonic materials.

cond-mat.mtrl-sci↗

arXiv2010

Densest local packing diversity. II. Application to three dimensions

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. 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. We additionally observe that the densest local packings are dominated by the spheres arranged with centers at precisely distance Rmin(N) from the fixed sphere's center.

cond-mat.stat-mech↗

arXiv2010

Duality Relations for the Classical Ground States of Soft-Matter Systems

Bounded interactions are particularly important in soft-matter systems, such as colloids, microemulsions, and polymers. We derive new duality relations for a class of soft potentials, including three-body and higher-order functions, that can be applied to ordered and disordered classical ground states. These duality relations link the energy of configurations associated with a real-space potential to the corresponding energy of the dual (Fourier-transformed) potential. We apply the duality relations by demonstrating how information about the classical ground states of short-ranged potentials can be used to draw new conclusions about the ground states of long-ranged potentials and vice versa. The duality relations also lead to bounds on the T=0 system energies in density intervals of phase coexistence. Additionally, we identify classes of "self-similar" potentials, for which one can relate low- and high-density ground-state energies. We analyze the ground state configurations and thermodynamic properties of a one-dimensional system previously thought to exhibit an infinite number of structural phase transitions and comment on the known ground states of purely repulsive monotonic potentials in the context of our duality relations.

cond-mat.stat-mech↗

arXiv2010

Jammed Hard-Particle Packings: From Kepler to Bernal and Beyond

This review describes the diversity of jammed configurations attainable by frictionless convex nonoverlapping (hard) particles in Euclidean spaces and for that purpose it stresses individual-packing geometric analysis. A fundamental feature of that diversity is the necessity to classify individual jammed configurations according to whether they are locally, collectively, or strictly jammed. Each of these categories contains a multitude of jammed configurations spanning a wide and (in the large system limit) continuous range of intensive properties, including packing fraction $ϕ$, mean contact number $Z$, and several scalar order metrics. Application of these analytical tools to spheres in three dimensions (an analog to the venerable Ising model) covers a myriad of jammed states, including maximally dense packings (as Kepler conjectured), low-density strictly-jammed tunneled crystals, and a substantial family of amorphous packings. With respect to the last of these, the current approach displaces the historically prominent but ambiguous idea of ``random close packing" (RCP) with the precise concept of ``maximally random jamming" (MRJ). This review also covers recent advances in understanding jammed packings of polydisperse sphere mixtures, as well as convex nonspherical particles, e.g., ellipsoids, ``superballs", and polyhedra. Because of their relevance to error-correcting codes and information theory, sphere packings in high-dimensional Euclidean spaces have been included as well. We also make some remarks about packings in (curved) non-Euclidean spaces. In closing this review, several basic open questions for future research to consider have been identified.

cond-mat.stat-mech↗

arXiv2010

Designer disordered materials with large complete photonic band gaps

We present designs of 2D isotropic, disordered photonic materials of arbitrary size with complete band gaps blocking all directions and polarizations. The designs with the largest gaps are obtained by a constrained optimization method that starts from a hyperuniform disordered point pattern, an array of points whose number variance within a spherical sampling window grows more slowly than the volume. We argue that hyperuniformity, combined with uniform local topology and short-range geometric order, can explain how complete photonic band gaps are possible without long-range translational order. We note the ramifications for electronic and phononic band gaps in disordered materials.

physics.optics↗

arXiv2010

Complete Band Gaps in 2D Photonic Quasicrystals

We introduce a novel optimization method to design the first examples of photonic quasicrystals with substantial, complete photonic band gaps (PBGs): that is, a range of frequencies over which electromagnetic wave propagation is forbidden for all directions and polarizations. The method can be applied to photonic quasicrystals with arbitrary rotational symmetry; here, we illustrate the results for 5- and 8-fold symmetric quasicrystals. The optimized band gaps are highly isotropic, which may offer advantages over photonic crystals for certain applications.

physics.optics↗

arXiv2010

Phase Behavior of Colloidal Superballs: Shape Interpolation from Spheres to Cubes

The phase behavior of hard superballs is examined using molecular dynamics within a deformable periodic simulation box. A superball's interior is defined by the inequality $|x|^{2q} + |y|^{2q} + |z|^{2q} \leq 1$, which provides a versatile family of convex particles ($q \geq 0.5$) with cube-like and octahedron-like shapes as well as concave particles ($q < 0.5$) with octahedron-like shapes. Here, we consider the convex case with a deformation parameter q between the sphere point (q = 1) and the cube (q = 1). We find that the asphericity plays a significant role in the extent of cubatic ordering of both the liquid and crystal phases. Calculation of the first few virial coefficients shows that superballs that are visually similar to cubes can have low-density equations of state closer to spheres than to cubes. Dense liquids of superballs display cubatic orientational order that extends over several particle lengths only for large q. Along the ordered, high-density equation of state, superballs with 1 < q < 3 exhibit clear evidence of a phase transition from a crystal state to a state with reduced long-ranged orientational order upon the reduction of density. For $q \geq 3$, long-ranged orientational order persists until the melting transition. The width of coexistence region between the liquid and ordered, high-density phase decreases with q up to q = 4.0. The structures of the high-density phases are examined using certain order parameters, distribution functions, and orientational correlation functions. We also find that a fixed simulation cell induces artificial phase transitions that are out of equilibrium. Current fabrication techniques allow for the synthesis of colloidal superballs, and thus the phase behavior of such systems can be investigated experimentally.

cond-mat.stat-mech↗

arXiv2009

Hyperuniformity in point patterns and two-phase random heterogeneous media

Hyperuniform point patterns are characterized by vanishing infinite wavelength density fluctuations and encompass all crystal structures, certain quasi-periodic systems, and special disordered point patterns. This article generalizes the notion of hyperuniformity to include also two-phase random heterogeneous media. Hyperuniform random media do not possess infinite-wavelength volume fraction fluctuations, implying that the variance in the local volume fraction in an observation window decays faster than the reciprocal window volume as the window size increases. For microstructures of impenetrable and penetrable spheres, we derive an upper bound on the asymptotic coefficient governing local volume fraction fluctuations in terms of the corresponding quantity describing the variance in the local number density (i.e., number variance). Extensive calculations of the asymptotic number variance coefficients are performed for a number of disordered (e.g., quasiperiodic tilings, classical stealth disordered ground states, and certain determinantal point processes), quasicrystal, and ordered (e.g., Bravais and non-Bravais periodic systems) hyperuniform structures in various Euclidean space dimensions, and our results are consistent with a quantitative order metric characterizing the degree of hyperuniformity. We also present corresponding estimates for the asymptotic local volume fraction coefficients for several lattice families. Our results have interesting implications for a certain problem in number theory.

cond-mat.stat-mech↗

arXiv2009

Novel Low-Temperature Behavior in Classical Many-Particle Systems

We show that classical many-particle systems interacting with certain soft pair interactions in two dimensions exhibit novel low-temperature behaviors. Ground states span from disordered to crystalline. At some densities, a large fraction of normal-mode frequencies vanish. Lattice ground-state configurations have more vanishing frequencies than disordered ground states at the same density and exhibit vanishing shear moduli. For the melting transition from a crystal, the thermal expansion coefficient is negative. These unusual results are attributed to the topography of the energy landscape.

cond-mat.stat-mech↗

arXiv2009

Dense sphere packings from optimized correlation functions

Elementary smooth functions (beyond contact) are employed to construct pair correlation functions that mimic jammed disordered sphere packings. Using the g2-invariant optimization method of Torquato and Stillinger [J. Phys. Chem. B 106, 8354, 2002], parameters in these functions are optimized under necessary realizability conditions to maximize the packing fraction phi and average number of contacts per sphere Z. A pair correlation function that incorporates the salient features of a disordered packing and that is smooth beyond contact is shown to permit a phi of 0.6850: this value represents a 45% reduction in the difference between the maximum for congruent hard spheres in three dimensions, pi/sqrt{18} ~ 0.7405, and 0.64, the approximate fraction associated with maximally random jammed (MRJ) packings in three dimensions. We show that, surprisingly, the continued addition of elementary functions consisting of smooth sinusoids decaying as r^{-4} permits packing fractions approaching pi/sqrt{18}. A translational order metric is used to discriminate between degrees of order in the packings presented. We find that to achieve higher packing fractions, the degree of order must increase, which is consistent with the results of a previous study [Torquato et al., Phys. Rev. Lett. 84, 2064, 2000].

cond-mat.stat-mech↗

arXiv2008

Point processes in arbitrary dimension from fermionic gases, random matrix theory, and number theory

It is well known that one can map certain properties of random matrices, fermionic gases, and zeros of the Riemann zeta function to a unique point process on the real line. Here we analytically provide exact generalizations of such a point process in d-dimensional Euclidean space for any d, which are special cases of determinantal processes. In particular, we obtain the n-particle correlation functions for any n, which completely specify the point processes. We also demonstrate that spin-polarized fermionic systems have these same n-particle correlation functions in each dimension. The point processes for any d are shown to be hyperuniform. The latter result implies that the pair correlation function tends to unity for large pair distances with a decay rate that is controlled by the power law r^[-(d+1)]. We graphically display one- and two-dimensional realizations of the point processes in order to vividly reveal their "repulsive" nature. Indeed, we show that the point processes can be characterized by an effective "hard-core" diameter that grows like the square root of d. The nearest-neighbor distribution functions for these point processes are also evaluated and rigorously bounded. Among other results, this analysis reveals that the probability of finding a large spherical cavity of radius r in dimension d behaves like a Poisson point process but in dimension d+1 for large r and finite d. We also show that as d increases, the point process behaves effectively like a sphere packing with a coverage fraction of space that is no denser than 1/2^d.

cond-mat.stat-mech↗

arXiv2008

Inverse Optimization Techniques for Targeted Self-Assembly

This article reviews recent inverse statistical-mechanical methodologies that we have devised to optimize interaction potentials in soft matter systems that correspond to stable "target" structures. We are interested in finding the interaction potential, not necessarily pairwise additive or spherically symmetric, that stabilizes a targeted many-body system by generally incorporating complete configurational information. Unlike previous work, our primary interest is in the possible many-body structures that may be generated, some of which may include interesting but known structures, while others may represent entirely new structural motifs. Soft matter systems, such as colloids and polymers, offer a versatile means of realizing the optimized interactions. It is shown that these inverse approaches hold great promise for controlling self-assembly to a degree that surpasses the less-than-optimal path that nature has provided. Indeed, we envision being able to "tailor" potentials that produce varying degrees of disorder, thus extending the traditional idea of self-assembly to incorporate both amorphous and crystalline structures as well as quasicrystals. The notion of tailoring potentials that correspond to targeted structures is motivated by the rich fundamental statistical-mechanical issues and questions offered by this fascinating inverse problem as well as our recent ability to identify structures that have optimal bulk properties or desirable performance characteristics. Recent results have already led to a deeper basic understanding of the mathematical relationship between the collective structural behavior of many-body systems and their interactions, as well as optimized potentials that enable self-assembly of ordered and disordered particle configurations with novel structural and bulk properties.

cond-mat.soft↗

arXiv2008

Negative thermal expansion in single-component systems with isotropic interactions

We have devised an isotropic interaction potential that gives rise to negative thermal expansion (NTE) behavior in equilibrium many-particle systems in both two and three dimensions over a wide temperature and pressure range (including zero pressure). An optimization procedure is used in order to find a potential that yields a strong NTE effect. A key feature of the potential that gives rise to this behavior is the softened interior of its basin of attraction. Although such anomalous behavior is well known in material systems with directional interactions (e.g., zirconium tungstate), to our knowledge this is the first time that NTE behavior has been established to occur in single-component many-particle systems for isotropic interactions. Using constant-pressure Monte Carlo simulations, we show that as the temperature is increased, the system exhibits negative, zero and then positive thermal expansion before melting (for both two- and three-dimensional systems). The behavior is explicitly compared to that of a Lennard-Jones system, which exhibits typical expansion upon heating for all temperatures and pressures.

cond-mat.mtrl-sci↗

arXiv2008

Negative Poisson's ratio materials via isotropic interactions

We show that under tension, a classical many-body system with only isotropic pair interactions in a crystalline state can, counterintutively, have a negative Poisson's ratio, or auxetic behavior. We derive the conditions under which the triangular lattice in two dimensions and lattices with cubic symmetry in three dimensions exhibit a negative Poisson's ratio. In the former case, the simple Lennard-Jones potential can give rise to auxetic behavior. In the latter case, negative Poisson's ratio can be exhibited even when the material is constrained to be elastically isotropic.

cond-mat.mtrl-sci↗

arXiv2008

Gaussian core model phase diagram and pair correlations in high Euclidean dimensions

The physical properties of a classical many-particle system with interactions given by a repulsive Gaussian pair potential are extended to arbitrarily high Euclidean dimensions. The goals of this paper are to characterize the behavior of the pair correlation function (pcf) in various density regimes and to understand the phase properties of the Gaussian core model (GCM) as parametrized by dimension d. To this end, we explore the fluid and crystalline solid phases. For the dilute regime of the fluid phase, a cluster expansion of the pcf in reciprocal temperature is presented, the coefficients of which may be evaluated analytically due to the nature of the Gaussian potential. We present preliminary results concerning the convergence properties of this expansion. The analytical cluster expansion is related to numerical approximations for the pcf in the dense fluid regime by utilizing hypernetted chain, Percus-Yevick, and mean-field closures to the Ornstein-Zernike equation. Based on the results of these comparisons, we provide evidence in support of a decorrelation principle for the GCM in high Euclidean dimensions. In the solid phase, we consider the behavior of the freezing temperature in the limit of zero density and show that it approaches zero itself in this limit for any d via a collective coordinate argument. Duality relations with respect to the energies of a lattice and its dual are then discussed, and these relations aid in the Maxwell double-tangent construction of phase coexistence regions between dual lattices based on lattice summation energies. The results from this analysis are used to draw conclusions about the ground-state structures of the GCM for a given dimension.

cond-mat.stat-mech↗

arXiv2008

Classical Disordered Ground States: Super-Ideal Gases, and Stealth and Equi-Luminous Materials

Using a collective coordinate numerical optimization procedure, we construct ground-state configurations of interacting particle systems in various space dimensions so that the scattering of radiation exactly matches a prescribed pattern for a set of wave vectors. We show that the constructed ground states are, counterintuitively, disordered (i.e., possess no long-range order) in the infinite-volume limit. We focus on three classes of configurations with unique radiation scattering characteristics: (i)``stealth'' materials, which are transparent to incident radiation at certain wavelengths; (ii)``super-ideal'' gases, which scatter radiation identically to that of an ensemble of ideal gas configurations for a selected set of wave vectors; and (iii)``equi-luminous'' materials, which scatter radiation equally intensely for a selected set of wave vectors. We find that ground-state configurations have an increased tendency to contain clusters of particles as one increases the prescribed luminosity. Limitations and consequences of this procedure are detailed.

cond-mat.stat-mech↗

arXiv2007

Tilings of space and superhomogeneous point processes

We consider the construction of point processes from tilings, with equal volume tiles, of d-dimensional Euclidean space. We show that one can generate, with simple algorithms ascribing one or more points to each tile, point processes which are "superhomogeneous'' (or "hyperuniform''), i.e., for which the structure factor S(k) vanishes when the wavenumber k tends to zero. The exponent of the leading small-k behavior depends in a simple manner on the nature of the correlation properties of the specific tiling and on the conservation of the mass moments of the tiles. Assigning one point to the center of mass of each tile gives the exponent γ=4 for any tiling in which the shapes and orientations of the tiles are short-range correlated. Smaller exponents, in the range 4-d<γ<4 (and thus always superhomogeneous for d\leq 4), may be obtained in the case that the latter quantities have long-range correlations. Assigning more than one point to each tile in an appropriate way, we show that one can obtain arbitrarily higher exponents in both cases. We illustrate our results with explicit constructions using known deterministic tilings, as well as some simple stochastic tilings for which we can calculate S(k) exactly. Our results provide, we believe, the first explicit analytical construction of point processes with γ> 4. Applications to condensed matter physics, and also to cosmology, are briefly discussed.

cond-mat.stat-mech↗

arXiv2007

Synthetic Diamond and Wurtzite Structures Self-Assemble with Isotropic Pair Interactions

Using inverse statistical-mechanical optimization techniques, we have discovered isotropic pair interaction potentials with strongly repulsive cores that cause the tetrahedrally coordinated diamond and wurtzite lattices to stabilize, as evidenced by lattice sums, phonon spectra, positive-energy defects, and self-assembly in classical molecular dynamics simulations. These results challenge conventional thinking that such open lattices can only be created via directional covalent interactions observed in nature. Thus, our discovery adds to fundamental understanding of the nature of the solid state by showing that isotropic interactions enable the self-assembly of open crystal structures with a broader range of coordination number than previously thought. Our work is important technologically because of its direct relevance generally to the science of self-assembly and specifically to photonic crystal fabrication.

cond-mat.soft↗