S. Torquato

On the design of 1-3 piezocomposites using topology optimization

We use a topology optimization method to design 1–3 piezocomposites with optimal performance characteristics for hydrophone applications. The performance characteristics we focus on are the hydrostatic charge coefficient , the hydrophone figure of merit , and the electromechanical coupling factor . The piezocomposite consists of piezoelectric rods embedded in an optimal polymer matrix. We use the topology optimization method to design the optimal (porous) matrix microstructure. When we design for maximum and , the optimal transversally isotopic matrix material has negative Poissons ratio in certain directions. When we design for maximum , the optimal matrix microstructure is layered and simple to build.

Chord‐length and free‐path distribution functions for many‐body systems

We study fundamental morphological descriptors of disordered media (e.g., heterogeneous materials, liquids, and amorphous solids): the chord‐length distribution function p(z) and the free‐path distribution function p(z,a). For concreteness, we will speak in the language of heterogeneous materials composed of two different materials or ‘‘phases.’’ The probability density function p(z) describes the distribution of chord lengths in the sample and is of great interest in stereology. For example, the first moment of p(z) is the ‘‘mean intercept length’’ or ‘‘mean chord length.’’ The chord‐length distribution function is of importance in transport phenomena and problems involving ‘‘discrete free paths’’ of point particles (e.g., Knudsen diffusion and radiative transport). The free‐path distribution function p(z,a) takes into account the finite size of a simple particle of radius a undergoing discrete free‐path motion in the heterogeneous material and we show that it is actually the chord‐length distribution fu...

Robust algorithm to generate a diverse class of dense disordered and ordered sphere packings via linear programming.

We have formulated the problem of generating dense packings of nonoverlapping, nontiling nonspherical particles within an adaptive fundamental cell subject to periodic boundary conditions as an optimization problem called the adaptive-shrinking cell (ASC) formulation [S. Torquato and Y. Jiao, Phys. Rev. E 80, 041104 (2009)]. Because the objective function and impenetrability constraints can be exactly linearized for sphere packings with a size distribution in d-dimensional Euclidean space R(d), it is most suitable and natural to solve the corresponding ASC optimization problem using sequential-linear-programming (SLP) techniques. We implement an SLP solution to produce robustly a wide spectrum of jammed sphere packings in R(d) for d=2, 3, 4, 5, and 6 with a diversity of disorder and densities up to the respective maximal densities. A novel feature of this deterministic algorithm is that it can produce a broad range of inherent structures (locally maximally dense and mechanically stable packings), besides the usual disordered ones (such as the maximally random jammed state), with very small computational cost compared to that of the best known packing algorithms by tuning the radius of the influence sphere. For example, in three dimensions, we show that it can produce with high probability a variety of strictly jammed packings with a packing density anywhere in the wide range [0.6, 0.7408...], where π/√18 = 0.7408... corresponds to the density of the densest packing. We also apply the algorithm to generate various disordered packings as well as the maximally dense packings for d=2, 4, 5, and 6. Our jammed sphere packings are characterized and compared to the corresponding packings generated by the well-known Lubachevsky-Stillinger (LS) molecular-dynamics packing algorithm. Compared to the LS procedure, our SLP protocol is able to ensure that the final packings are truly jammed, produces disordered jammed packings with anomalously low densities, and is appreciably more robust and computationally faster at generating maximally dense packings, especially as the space dimension increases.

Equi-g(r) sequence of systems derived from the square-well potential

We introduce the idea of an “equi-g(r) sequence.” This consists of a series of equilibrium many-body systems which have different number densities ρ but share, at a given temperature, the same form of pair correlation function, termed “target g(r).” Each system is defined by a pair potential indexed by ρ as in uρ(r). It is shown that for such a sequence a terminal density ρ⋆ exists, beyond which no physically realizable system can be found. As an illustration we derive explicit values of ρ⋆ for target g(r) that is based on a square-well potential in the limit ρ→0. Possible application of this terminal phenomenon to the investigation into limiting amorphous packing structures of hard spheres is proposed. Virial expansions of uρ(r) and pressure are carried out and compared with the corresponding expressions for imperfect gas. The behaviors of uρ(r) and pressure close to ρ=ρ⋆ are examined as well, and associated exponents extracted when they exist. The distinction between equi-g(r) sequence and the related, ...

Local volume fraction fluctuations in periodic heterogeneous media

Although the volume fraction is a constant for statistically homogeneous media, on a spatially local level it fluctuates and depends on the observation window size. In this article, we develop exact analytical expressions for the full local-volume fraction distribution function of periodic arrangements of rods, rectangles, and cubes in a matrix. These formulas depend on the inclusion density and window size.

New classes of non-crystalline photonic band gap materials

Due to their ability to control the most fundamental properties of light, photonic band gap (PBG) materials open a new frontier in both basic science and technology [1,2]. Until now, the only materials known to have complete photonic band gaps were photonic crystals, periodic structures. We show that there exists a more general class of systems, called hyperuniform photonic structures, which exhibit large and complete photonic band gaps. The common feature of hyperuniform photonic structures considered here is that they are derived from hyperuniform point patterns [3]. Hyperuniform point patterns are those whose number variance within a spherical sampling window of radius R (in d dimensions) grows more slowly than the window volume for large R, i.e., 〈N<inf>R</inf>²〉 − 〈N<inf>R</inf>〉² ∼ R^p, where p ≪ d. This classification includes all crystals and quasicrystals [4], as well as a special subset of disordered structures.

On the Design of Hydrophones made as 1–3 Piezoelectrics

We compare two different approaches to the design of 1–3 piezocomposites with optimal performance characteristic for hydrophone applications. The performance characteristics we focus is the hydrostatic charge coefficient d h (*) . The piezocomposite consists of piezoelectric rods embedded in an optimal polymer matrix. In the first approach, we allow the material parameters of the polymer matrix to be free design variables and we find the polymer parameters which give the optimal hydrophone performance. In the second approach, we use the topology optimization method to design the optimal (porous) matrix microstructure. Using the former approach, we find theoretical bounds for hydrophone behavior and using the latter, we design practically realizable but less efficient hydrophones. For both approaches, the optimal transversally isotropic matrix material has negative Poisson’s ratio in certain directions.