Platonic crystal with low-frequency locally resonant snail structures. Wave trapping, transmission amplification and shielding

We propose a new type of platonic crystal. The proposed microstructured plate includes snail resonators with low-frequency resonant vibrations. The particular dynamic effect of the resonators are highlighted by a comparative analysis of dispersion properties of homo- geneous and perforated plates. Analytical and numerical estimates of classes of standing waves are given and the analysis on a macrocell shows the possibility to obtain localization, wave trapping and edge waves. Applications include transmission amplification within two plates separated by a small ligament. Finally we proposed a design procedure to suppress low frequency flexural vibration in an elongated plate implementing a by-pass system re- routing waves within the mechanical system.Comment: 11 figures (20 files

Conduction in jammed systems of tetrahedra

Control of transport processes in composite microstructures is critical to the development of high performance functional materials for a variety of energy storage applications. The fundamental process of conduction and its control through the manipulation of granular composite attributes (e.g., grain shape) are the subject of this work. We show that athermally jammed packings of tetrahedra with ultra-short range order exhibit fundamentally different pathways for conduction than those in dense sphere packings. Highly resistive granular constrictions and few face-face contacts between grains result in short-range distortions from the mean temperature field. As a consequence, 'granular' or differential effective medium theory predicts the conductivity of this media within 10% at the jamming point; in contrast, strong enhancement of transport near interparticle contacts in packed-sphere composites results in conductivity divergence at the jamming onset. The results are expected to be particularly relevant to the development of nanomaterials, where nanoparticle building blocks can exhibit a variety of faceted shapes.Comment: 9 pages, 10 figure

Trapped Modes and Steered Dirac Cones in Platonic Crystals

This paper discusses the properties of flexural waves obeying the biharmonic equation, propagating in a thin plate pinned at doubly-periodic sets of points. The emphases are on the properties of dispersion surfaces having the Dirac cone topology, and on the related topic of trapped modes in plates with a finite set (cluster) of pinned points. The Dirac cone topologies we exhibit have at least two cones touching at a point in the reciprocal lattice, augmented by another band passing through the point. We show that the Dirac cones can be steered along symmetry lines in the Brillouin zone by varying the aspect ratio of rectangular lattices of pins, and that, as the cones are moved, the involved band surfaces tilt. We link Dirac points with a parabolic profile in their neighbourhood, and the characteristic of this parabolic profile decides the direction of propagation of the trapped mode in finite clusters.Comment: 21 pages, 12 figure

Berezinskii-Kosterlitz-Thouless Type Scenario in Molecular Spin Liquid AACr2_2O4_4

The spin relaxation in chromium spinel oxides $A$Cr$_{2}$O$_{4}$ ($A=$ Mg, Zn, Cd) is investigated in the paramagnetic regime by electron spin resonance (ESR). The temperature dependence of the ESR linewidth indicates an unconventional spin-relaxation behavior, similar to spin-spin relaxation in the two-dimensional (2D) chromium-oxide triangular lattice antiferromagnets. The data can be described in terms of a generalized Berezinskii-Kosterlitz-Thouless (BKT) type scenario for 2D systems with additional internal symmetries. Based on the characteristic exponents obtained from the evaluation of the ESR linewidth, short-range order with a hidden internal symmetry is suggested.Comment: 7 pages, 4 figure

Basic Understanding of Condensed Phases of Matter via Packing Models

Packing problems have been a source of fascination for millenia and their study has produced a rich literature that spans numerous disciplines. Investigations of hard-particle packing models have provided basic insights into the structure and bulk properties of condensed phases of matter, including low-temperature states (e.g., molecular and colloidal liquids, crystals and glasses), multiphase heterogeneous media, granular media, and biological systems. The densest packings are of great interest in pure mathematics, including discrete geometry and number theory. This perspective reviews pertinent theoretical and computational literature concerning the equilibrium, metastable and nonequilibrium packings of hard-particle packings in various Euclidean space dimensions. In the case of jammed packings, emphasis will be placed on the "geometric-structure" approach, which provides a powerful and unified means to quantitatively characterize individual packings via jamming categories and "order" maps. It incorporates extremal jammed states, including the densest packings, maximally random jammed states, and lowest-density jammed structures. Packings of identical spheres, spheres with a size distribution, and nonspherical particles are also surveyed. We close this review by identifying challenges and open questions for future research.Comment: 33 pages, 20 figures, Invited "Perspective" submitted to the Journal of Chemical Physics. arXiv admin note: text overlap with arXiv:1008.298

Exact Constructions of a Family of Dense Periodic Packings of Tetrahedra

The determination of the densest packings of regular tetrahedra (one of the five Platonic solids) is attracting great attention as evidenced by the rapid pace at which packing records are being broken and the fascinating packing structures that have emerged. Here we provide the most general analytical formulation to date to construct dense periodic packings of tetrahedra with four particles per fundamental cell. This analysis results in six-parameter family of dense tetrahedron packings that includes as special cases recently discovered "dimer" packings of tetrahedra, including the densest known packings with density $\phi= 4000/4671 = 0.856347...$. This study strongly suggests that the latter set of packings are the densest among all packings with a four-particle basis. Whether they are the densest packings of tetrahedra among all packings is an open question, but we offer remarks about this issue. Moreover, we describe a procedure that provides estimates of upper bounds on the maximal density of tetrahedron packings, which could aid in assessing the packing efficiency of candidate dense packings.Comment: It contains 25 pages, 5 figures

The Geometry of Slow Structural Fluctuations in a Supercooled Binary Alloy

The liquid structure of a glass-forming binary alloy is studied using molecular dynamics simulations. The analysis combines common neighbour analysis with the geometrical approach of Frank and Kasper to establish that the supercooled liquid contains extended clusters characterised by the same short range order as the crystal. Fluctuations in these clusters exhibit strong correlations with fluctuations in the inherent structure energy. The steep increase in the heat capacity on cooling is, thus, directly coupled to the growing fluctuations of the Frank-Kasper clusters. The relaxation of particles in the clusters dominates the slow tail of the self-intermediate scattering function

A Collection of Challenging Optimization Problems in Science, Engineering and Economics

Function optimization and finding simultaneous solutions of a system of nonlinear equations (SNE) are two closely related and important optimization problems. However, unlike in the case of function optimization in which one is required to find the global minimum and sometimes local minima, a database of challenging SNEs where one is required to find stationary points (extrama and saddle points) is not readily available. In this article, we initiate building such a database of important SNE (which also includes related function optimization problems), arising from Science, Engineering and Economics. After providing a short review of the most commonly used mathematical and computational approaches to find solutions of such systems, we provide a preliminary list of challenging problems by writing the Mathematical formulation down, briefly explaning the origin and importance of the problem and giving a short account on the currently known results, for each of the problems. We anticipate that this database will not only help benchmarking novel numerical methods for solving SNEs and function optimization problems but also will help advancing the corresponding research areas.Comment: Accepted as an invited contribution to the special session on Evolutionary Computation for Nonlinear Equation Systems at the 2015 IEEE Congress on Evolutionary Computation (at Sendai International Center, Sendai, Japan, from 25th to 28th May, 2015.