Aleksandar Donev

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.

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.

Jamming in hard sphere and disk packings

Hard-particle packings have provided a rich source of outstanding theoretical problems and served as useful starting points to model the structure of granular media, liquids, living cells, glasses, and random media. The nature of “jammed” hard-particle packings is a current subject of keen interest. Elsewhere, we introduced rigorous and efficient linear-programming algorithms to assess whether a hard-sphere packing is locally, collectively, or strictly jammed, as defined by Torquato and Stillinger [J. Phys. Chem. B 105, 11849 (2001)]. One algorithm applies to ideal packings in which particles form perfect contacts. Another algorithm treats the case of jamming in packings with significant interparticle gaps. We have applied these algorithms to test jamming categories of ordered lattices as well as random packings of circular disks and spheres under periodic boundary conditions. The random packings were produced computationally with a variety of packing generation algorithms, all of which should, in princip...

Underconstrained jammed packings of nonspherical hard particles: ellipses and ellipsoids.

Continuing on recent computational and experimental work on jammed packings of hard ellipsoids [Donev, Science 303, 990 (2004)] we consider jamming in packings of smooth strictly convex nonspherical hard particles. We explain why an isocounting conjecture, which states that for large disordered jammed packings the average contact number per particle is twice the number of degrees of freedom per particle (Z[over]=2d{f}) , does not apply to nonspherical particles. We develop first- and second-order conditions for jamming and demonstrate that packings of nonspherical particles can be jammed even though they are underconstrained (hypoconstrained, Z[over]<2d{f}). We apply an algorithm using these conditions to computer-generated hypoconstrained ellipsoid and ellipse packings and demonstrate that our algorithm does produce jammed packings, even close to the sphere point. We also consider packings that are nearly jammed and draw connections to packings of deformable (but stiff) particles. Finally, we consider the jamming conditions for nearly spherical particles and explain quantitatively the behavior we observe in the vicinity of the sphere point.

Optimal design of manufacturable three-dimensional composites with multifunctional characteristics

We present an optimization method to design three-dimensional composite microstructures with multifunctional characteristics. To illustrate the fascinating types of microstructures that can arise in multifunctional optimization, we apply our methodology to the study the simultaneous transport of heat and electricity in three-dimensional, two-phase composites. We assume that phase 1 has a high thermal conductivity but low electrical conductivity and phase 2 has a low thermal conductivity but high electrical conductivity. The objective functions consist of different combinations of the dimensionless effective thermal and electrical conductivities. When the sum of the effective thermal and electrical conductivities is maximized, we find that the optimal three-dimensional microstructures are triply periodic bicontinuous composites with interfaces that are the Schwartz primitive (P) and diamond (D) minimal surfaces. Maximizing the effective thermal conductivity and minimizing the effective electrical conductiv...

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).

Staggered schemes for fluctuating hydrodynamics

We develop numerical schemes for solving the isothermal compressible and incompressible equations of fluctuating hydrodynamics on a grid with staggered momenta. We develop a second-order accurate spatial discretization of the diffusive, advective, and stochastic fluxes that satisfies a discrete fluctuation-dissipation balance and construct temporal discretizations that are at least second-order accurate in time deterministically and in a weak sense. Specifically, the methods reproduce the correct equilibrium covariances of the fluctuating fields to the third (compressible) and second (incompressible) orders in the time step, as we verify numerically. We apply our techniques to model recent experimental measurements of giant fluctuations in diffusively mixing fluids in a microgravity environment [A. Vailati et al., Nat. Comm., 2 (2011), 290]. Numerical results for the static spectrum of nonequilibrium concentration fluctuations are in excellent agreement between the compressible and incompressible simulati...

A First-Passage Kinetic Monte Carlo algorithm for complex diffusion-reaction systems

We develop an asynchronous event-driven First-Passage Kinetic Monte Carlo (FPKMC) algorithm for continuous time and space systems involving multiple diffusing and reacting species of spherical particles in two and three dimensions. The FPKMC algorithm presented here is based on the method introduced in Oppelstrup et al. [10] and is implemented in a robust and flexible framework. Unlike standard KMC algorithms such as the n-fold algorithm, FPKMC is most efficient at low densities where it replaces the many small hops needed for reactants to find each other with large first-passage hops sampled from exact time-dependent Greens functions, without sacrificing accuracy. We describe in detail the key components of the algorithm, including the event-loop and the sampling of first-passage probability distributions, and demonstrate the accuracy of the new method. We apply the FPKMC algorithm to the challenging problem of simulation of long-term irradiation of metals, relevant to the performance and aging of nuclear materials in current and future nuclear power plants. The problem of radiation damage spans many decades of time-scales, from picosecond spikes caused by primary cascades, to years of slow damage annealing and microstructure evolution. Our implementation of the FPKMC algorithm has been able to simulate the irradiation of a metal sample for durations that are orders of magnitude longer than any previous simulations using the standard Object KMC or more recent asynchronous algorithms.

First-passage kinetic Monte Carlo method.

We present an efficient method for Monte Carlo simulations of diffusion-reaction processes. Introduced by us in a previous paper [Phys. Rev. Lett. 97, 230602 (2006)], our algorithm skips the traditional small diffusion hops and propagates the diffusing particles over long distances through a sequence of superhops, one particle at a time. By partitioning the simulation space into nonoverlapping protecting domains each containing only one or two particles, the algorithm factorizes the N -body problem of collisions among multiple Brownian particles into a set of much simpler single-body and two-body problems. Efficient propagation of particles inside their protective domains is enabled through the use of time-dependent Greens functions (propagators) obtained as solutions for the first-passage statistics of random walks. The resulting Monte Carlo algorithm is event-driven and asynchronous; each Brownian particle propagates inside its own protective domain and on its own time clock. The algorithm reproduces the statistics of the underlying Monte Carlo model exactly. Extensive numerical examples demonstrate that for an important class of diffusion-reaction models the algorithm is efficient at low particle densities, where other existing algorithms slow down severely.

Minimal surfaces and multifunctionality

Triply periodic minimal surfaces are objects of great interest to physical scientists, biologists and mathematicians. It has recently been shown that triply periodic two-phase bicontinuous composites with interfaces that are the Schwartz primitive (P) and diamond (D) minimal surfaces are not only geometrically extremal but extremal for simultaneous transport of heat and electricity. More importantly, here we further establish the multifunctionality of such two-phase systems by showing that they are also extremal when a competition is set up between the effective bulk modulus and the electrical (or thermal) conductivity of the composite. The implications of our findings for materials science and biology, which provides the ultimate multifunctional materials, are discussed.