Lattice Of Spheres Research Articles

Acoustic performance of screens designed to act as metamaterials

Metamaterials are, nowadays, a mature field of research in acoustic and many applications with metamaterials have been developed in recent years. In metamaterials, the attenuation of sound is due to the interaction of the sound waves with their geometrically regular structures. The frequency at which the attenuation occurs is calculated with Braggs' law. This work presents sound attenuation measurements of structures made with metamaterials carried out in an anechoic chamber. The arrangements considered are both two- dimensional and three-dimensional: the 2D arrangement was obtained with a regular distribution of linear elements made with bars with diameters of 9, 15, and 20 mm; the 3D arrangement was obtained with a regular distribution of a lattice of spheres with a diameter of 23 mm. The results of the acoustic measurements are reported in terms of insertion loss in the range from 1000 to 10 000 Hz.

Weak-form homogenization of two and three-dimensional fluid acoustical systems.

A one-dimensional weak-form homogenization method [Muhlestein, J. Acoust. Soc. Am. 147(5), 3584-3593 (2020)] is extended to two and three-dimensional for quasi-static fluid systems. This homogenization approach uses a local multiple-scales approximation to estimate the acoustical fields within a representative volume element, substitutes these approximations into a weak formulation of the mechanics, and then globally homogenizes the system by averaging the integrand of the weak-form integral. An important consequence of including more spatial dimensions is that the local particle velocity does not approach a uniform macroscopic particle velocity. Instead, the effective material properties are used to describe the behavior of the mean particle velocity. A localization tensor may be used to convert from the mean particle velocity to the local particle velocity. The generalized homogenization method is then applied to two special cases. The first case is stratified media, chosen because it has an exact analytical solution. The second case is a cubic lattice of spheres, which has a benchmark solution to compare with. This second case utilizes finite element software to provide estimates of the effective mass density. Finally, three further generalizations to the homogenization method, including extension to finite frequency values, complex media, and elasticity, are briefly discussed.

A Dosimetric Analysis of Single Fraction Lattice Radiotherapy as a Boost for the Palliative Treatment of Bulky Head and Neck Tumors

<h3>Purpose/Objective(s)</h3> This study investigates the dosimetry achieved in VMAT/IMRT-planned lattice radiotherapy (LRT) for bulky, palliative head and neck tumors. LRT allows for the delivery of higher radiation doses without dramatically increasing toxicity by using high-dose spheres within the tumor while directing high dose away from OAR's. Our objective was to analyze LRT using 3 dose levels as a boost for palliative treatment of bulky head and neck tumors. The clinical goal of LRT boost would be to increase the effectiveness of treatment while minimizing the burden of care for patients requiring a quick palliation of symptoms associated with bulky tumors. <h3>Materials/Methods</h3> Six bulky palliative head and neck cases were used for this study. A 3D lattice of 0.75 cm spheres spaced 0.85 cm apart was applied to the planning scan and restricted to the area inside the lattice inward contour (GTV was contracted by 0.5 mm). Each case was planned with three different dose prescriptions (15 Gy, 20 Gy and 25 Gy in a single fraction) to the lattice of spheres. In addition, a standard IMRT plan was also created to deliver a uniform 20 Gy in 5 fractions to a defined PTV. The margins (CTV + PTV expansions) used were between 0.5 to 1 cm. A total of 7 plans were created for each case, which included the three single fraction doses, the five fractions of standard treatment, and a combination of the single and five fraction plans. <h3>Results</h3> The average planned tumor volume was 209.77 mL (range 137.80 – 411.01). The number of spheres used in the LRT plan ranged from 13 to 41. The median maximum dose (Dmax) and dose statistics achieved are listed in Table 1. The median ipsilateral carotid dose was high for the 25 Gy (single fraction) plan, so our results suggest that a neoadjuvant LRT boost with of a single fraction of 20 Gy would likely be safe. <h3>Conclusion</h3> Lattice radiotherapy is a viable treatment option for bulky head and neck cases treated with palliative intent. We were able to achieve acceptable dosimetric plans in both the 15 and 20 Gy boosts (single fraction) followed by 20 Gy in 5 fractions, a combination that would be expected to provide more favorable tumor control and symptom relief than traditional palliative regimens of 20-30 Gy. A clinical application with prospective dose escalation trial is under consideration.

Defining the pressures of a fluid in a nanoporous, heterogeneous medium

We describe the thermodynamic state of a single-phase fluid confined to a porous medium with Hill’s thermodynamics of small systems, also known as nanothermodynamics. This way of defining small system thermodynamics, with a separate set of control variables, may be useful for the study of transport in non-deformable porous media, where presently no consensus exists on pressure computations. For a confined fluid, we observe that there are two pressures, the integral and the differential pressures. We use molecular simulations to investigate and confirm the nanothermodynamic relations for a representative elementary volume (REV). For a model system of a single-phase fluid in a face-centered cubic lattice of solid spheres of varying porosity, we calculate the fluid density, fluid-solid surface tension, replica energy, integral pressure, entropy, and internal energy.

Local Thermodynamic Description of Isothermal Single-Phase Flow in Simple Porous Media

Darcy’s law for porous media transport is given a new local thermodynamic basis in terms of the grand potential of confined fluids. The local effective pressure gradient is determined using non-equilibrium molecular dynamics, and the hydraulic conductivity and permeability are investigated. The transport coefficients are determined for single-phase flow in face-centered cubic lattices of solid spheres. The porosity changed from that in the closest packing of spheres to near unity in a pure fluid, while the fluid mass density varied from that of a dilute gas to a dense liquid. The permeability varied between 5.7 times {10^{-20}} hbox {m}^2 and 5.5 times {10^{-17}} hbox {m}^2, showing a porosity-dependent Klinkenberg effect. Both transport coefficients depended on the average fluid mass density and porosity but in different ways. These results set the stage for a non-equilibrium thermodynamic investigation of coupled transport of multi-phase fluids in complex media.

Trade‐off Lattice Site Occupancy Engineering Strategy for Near‐Infrared Phosphors with Ultrabroad and Tunable Emission

AbstractBroadband near‐infrared (NIR) luminescent materials have received notable attention due to their distinct photophysical properties for designing NIR light‐emitting diodes (NIR LEDs). Here, a series of Ca3−xLuxGa2+xGe3−xO12:Cr3+ (CLGGG:Cr3+) (x = 0–1) NIR‐emitting garnet phosphors with an emission range of 660–1200 nm are successfully developed and their lattice parameters are structurally analyzed. Upon 460 nm blue light excitation, the NIR phosphors exhibit both a substantial spectral broadening (FWHM: 129→267 nm) and a redshift of 37 nm (766→803 nm) with cosubstitution of [Lu3+–Ga3+] pairs for [Ca2+–Ge4+] sites. Furthermore, their luminescence thermal stability is substantially improved, maintaining ≈90% of the original photoluminescence intensity at 150 °C, owing to shrinkage of the second coordination sphere and rigid lattice, which are strongly associated with Cr3+ trade‐off occupancy and local structure evolutions. The relation between the trade‐off site occupation of Cr3+ in GaO6/CaO8 polyhedrons and the NIR emission is also clarified by evaluating the decay and electron paramagnetic resonance behavior of Cr3+ at different sites. The broadband NIR phosphors investigated here can serve as auspicious luminescent converters for phosphor‐converted NIR LEDs and can provide an inspiring platform for future studies.

Magnetic Microparticles: Collective Transport of Magnetic Microparticles at a Fluid Interface through Dynamic Self‐Assembled Lattices (Adv. Funct. Mater. 50/2020)

In article number 2002206, Fernando Martínez-Pedrero, Carles Calero, and co-workers present an easily implementable strategy wherein small microparticles adsorbed at a fluid interface are transported through a lattice of larger spheres, in a practically deterministic motion, by periodically inverting the vertical component of a precessing field.

Optical method for measuring the volume fraction of granular media: Application to faced-centered cubic lattices of monodisperse spheres.

In order to understand the dynamics of granular flows, one must have knowledge about the solid volume fraction. However, its reliable experimental estimation is still a challenging task. Here, we present the application of a stochastic-optical method (SOM) [L. Sarno et al., Granul. Matter 18, 80 (2016)10.1007/s10035-016-0676-3] to an array of spheres arranged according to faced-centered cubic lattices, where spheres' locations are known a priori. The purpose of this study is to test the robustness of the image binarization algorithm, introduced in the SOM for the indirect estimation of the near-wall volume fraction through an optically measurable quantity, defined as two-dimensional volume fraction. A comprehensive range of volume fractions and illumination conditions are numerically and experimentally investigated. The proposed binarization algorithm is found to yield reasonably accurate estimations of the two-dimensional volume fraction with a root-mean-square error smaller than 0.03 for all investigated illumination conditions. A slightly worse performance is observed for samples with relatively low volume fractions (<0.3), where the binarization algorithm occasionally cannot identify the surface elements in the second and third layers of the regular lattice.

Chaotic mixing in crystalline granular media

We study the Lagrangian kinematics of steady three-dimensional Stokes flow over simple cubic (SC) and body-centred cubic (BCC) lattices of close-packed spheres, and uncover the mechanisms governing chaotic mixing in these crystalline structures. Due to the cusp-shaped sphere contacts, the topology of the skin friction field is fundamentally different to that of continuous (non-granular) media, such as open pore networks, with significant implications for fluid mixing. Weak symmetry breaking of the flow orientation with respect to the lattice symmetries imparts a transition from regular to strong chaotic mixing in the BCC lattice, whereas the SC lattice only exhibits weak mixing. Whilst the SC and BCC lattices posses the same symmetry point group, these differences are explained in terms of their space groups. This insight is used to develop accurate predictions of the Lyapunov exponent distribution over the parameter space of mean flow orientation. These results point to a general theory of mixing and dispersion based upon the inherent symmetries of arbitrary crystalline structures.

Characterization of maximally random jammed sphere packings. III. Transport and electromagnetic properties via correlation functions.

In the first two papers of this series, we characterized the structure of maximally random jammed (MRJ) sphere packings across length scales by computing a variety of different correlation functions, spectral functions, hole probabilities, and local density fluctuations. From the remarkable structural features of the MRJ packings, especially its disordered hyperuniformity, exceptional physical properties can be expected. Here we employ these structural descriptors to estimate effective transport and electromagnetic properties via rigorous bounds, exact expansions, and accurate analytical approximation formulas. These property formulas include interfacial bounds as well as universal scaling laws for the mean survival time and the fluid permeability. We also estimate the principal relaxation time associated with Brownian motion among perfectly absorbing traps. For the propagation of electromagnetic waves in the long-wavelength limit, we show that a dispersion of dielectric MRJ spheres within a matrix of another dielectric material forms, to a very good approximation, a dissipationless disordered and isotropic two-phase medium for any phase dielectric contrast ratio. We compare the effective properties of the MRJ sphere packings to those of overlapping spheres, equilibrium hard-sphere packings, and lattices of hard spheres. Moreover, we generalize results to micro- and macroscopically anisotropic packings of spheroids with tensorial effective properties. The analytic bounds predict the qualitative trend in the physical properties associated with these structures, which provides guidance to more time-consuming simulations and experiments. They especially provide impetus for experiments to design materials with unique bulk properties resulting from hyperuniformity, including structural-color and color-sensing applications.

Space-Group Symmetries Generate Chaotic Fluid Advection in Crystalline Granular Media.

The classical connection between symmetry breaking and the onset of chaos in dynamical systems harks back to the seminal theory of Noether [Transp. Theory Statist. Phys. 1, 186 (1918)10.1080/00411457108231446]. We study the Lagrangian kinematics of steady 3D Stokes flow through simple cubic and body-centered cubic (bcc) crystalline lattices of close-packed spheres, and uncover an important exception. While breaking of point-group symmetries is a necessary condition for chaotic mixing in both lattices, a further space-group (glide) symmetry of the bcc lattice generates a transition from globally regular to globally chaotic dynamics. This finding provides new insights into chaotic mixing in porous media and has significant implications for understanding the impact of symmetries upon generic dynamical systems.

Symmetry-guaranteed nodal-line semimetals in an fcc lattice

We demonstrate theoretically that nodal-line semimetals (NLSs) can be realized in an fcc lattice with orbitals belonging to the same irreducible representation, such as ${{p}_{x},{p}_{y},{p}_{z}}$ or ${{d}_{xy},{d}_{yz},{d}_{zx}}$ orbitals on every lattice site. The three orbitals are divided into two subgroups in terms of the parity with respect to the mirror reflections on high-symmetry planes of the fcc lattice, which, with rotation symmetry, endows symmetry-guaranteed NL passing through $W$ points in the Brillouin zone. Depending on the parameters, there also appears an accidental NL around the $\mathrm{\ensuremath{\Gamma}}$ point. We notice that the symmetry-guaranteed NL addressed in the present work can be found in band structures of elemental solids taking the fcc structure, such as Cu, Ag, Au, In, Ga, etc., as well as opal, which is an fcc photonic crystal of ${\mathrm{SiO}}_{2}$ spheres. Furthermore, we clarify that the fcc lattice of Si spheres exhibits a NL in a frequency band where no other photonic band exists, which provides a unique platform to realize topological NLSs under intensive search, and can be explored for achieving slow light.

Light diffraction features in an ordered monolayer of spheres

The structure and optical diffraction properties of monolayers of monodisperse spheres crystallized on transparent dielectric substrates are studied. Two types of diffraction phenomena are considered: surface light diffraction on the lattice of spheres and waveguide resonances in the monolayer plane. For experimental study of these phenomena, optical retroreflection and transmission spectra are measured as functions of the light incidence angle and azimuthal orientation of the incidence plane. The monolayer structures determined by scanning electron microscopy and light diffraction methods are in quantitative agreement. It is concluded that one-dimensional Fraunhofer diffraction is applicable to describe surface diffraction in the hexagonal lattice of spheres. In the case of oblique light incidence, anisotropy of diffraction and transmission spectra depending on the light incidence plane orientation with respect to the sphere lattice and linear polarization of incident light is detected. Waveguide resonances of the planar two-dimensional photonic crystal are approximated within the light diffraction model in the “empty” hexagonal lattice. The best approximation of the waveguide resonance dispersion is achieved using the effective refractive index, depending on the wavelength. Surface diffraction suppression by waveguide resonances of the photonic crystal is demonstrated. Surface diffraction orders are identified as diffraction at singular points of the Brillouin zone of the planar twodimensional photonic crystal.

On elastic waves in granular assemblies: From a continuumnization viewpoint

The characteristics of elastic waves in granular assemblies of spherical particles, which translate and spin, are analyzed by a continuumnization technique. The continuumnized governing equations for equal spheres in three-dimensional lattices are first obtained, accompanied by auxiliary tensors corresponding to the macroscopic material properties. The property tensors, i.e., the stiffness, coupling, and spinning tensors, given in neat analytical form, are solely determined by the micro-mechanics (the normal and tangential spring constants) and the microstructures of the lattices (the unit vectors connecting the centers of neighboring particles). The general formulas are then applied for two types of closest packed lattices. Theoretical wave velocities for p- and s-waves in such lattices are derived, together with the frequency for the pure spin mode and the group velocity for the spin motion coupled with s-waves. Two interesting findings of the physics of lattices of spheres are revealed: the s-wave coupled with spin motion in the lattices can be understood as acoustic and optical branches of waves in the diatomic model in solid-state physics; the anisotropic closest packed lattices would degenerate into a special isotropic state with zero Poisson’s ratio and a constant velocity ratio of 2 between p- and s-waves. How the wave velocities and the anisotropy are influenced by the spring constant ratios are also analyzed. The theoretical results obtained can be used for quantitative verification and for parameter calibration of discrete-element-method simulations using spherical particles.

Stimulated Brillouin scattering in metamaterials

We compute the SBS gain for a metamaterial comprising a cubic lattice of dielectric spheres suspended in a background dielectric material. Theoretical methods are presented to calculate the optical, acoustic, and opto-acoustic parameters that describe the SBS properties of the material at long wavelengths. Using the electromagnetic and strain energy densities we accurately characterise the optical and acoustic properties of the metamaterial. From a combination of energy density methods and perturbation theory, we recover the appropriate terms of the photoelastic tensor for the metamaterial. We demonstrate that electrostriction is not necessarily the dominant mechanism in the enhancement and suppression of the SBS gain coefficient in a metamaterial, and that other parameters, such as the Brillouin linewidth, can dominate instead. Examples are presented that exhibit an order of magnitude enhancement in the SBS gain as well as perfect suppression.

Decorated granular layers for impact decimation

We present dynamical simulations and simple mechanics arguments to propose a system of stacked blocks of square lattices of elastic spheres that can be used to decimate an incident impulse. Mass mismatch between adjacent blocks is accomplished by making the sphere radius in the upper block twice that of the lower block. The system decimates impact energies by converting the initial impulse into two solitary waves and then progressively into many smaller amplitude solitary waves. We also show that near perfect impact decimation capability can be realized with increased mass mismatch between adjacent blocks by creating sandwiched structures in which a block with smaller density spheres is surrounded on both sides with blocks of denser spheres. The proposed systems are expected to be scalable down to spheres of \(\sim \)100 nm and work for solid and hollow spheres.

Direct Observation on Spin-Coating Process of PS-b-P2VP Thin Films

We studied the structural development of symmetric poly(styrene-b-2-vinylpyridine) (PS-b-P2VP) block copolymers during spin-coating using in situ grazing incidence small-angle X-ray scattering (GISAXS) measurements. During the spin-coating process, after the formation of the micelles in dilute solution, the selective solvent induced two kinds of the morphological transition. First, the disordered spherical micelles were transformed into a BCC lattice of spheres of which the (110) plane was oriented perpendicularly to the substrate surface. Second, further evaporation induced a transition from spheres on the BCC lattice into cylindrical structures. The orientation of the cylinders perpendicular to the substrate surface was induced by solvent convection perpendicular to the substrate, which occurs during rapid solvent evaporation. After this transition, vitrification of PS and P2VP prevented any further transition from cylinders to the more thermodynamically stable lamellar structures, as are generally obse...

Tic-Tac-Toe Binary Lattices from the Interfacial Self-Assembly of Branched and Spherical Nanocrystals

The self-organization of nanocrystals has proven to be a versatile route to achieve increasingly sophisticated structures of materials, where the shape and properties of individual particles impact the final functionalities. Recent works have addressed this topic by combining various shapes to achieve more complex arrangements of particles than are possible in single-component samples. However, the ability to create intricate architectures over large regions by exploiting the shape of multiply branched nanocrystals to host a second component remains unexplored. Here, we show how the concave shape of a branched nanocrystal, the so-called octapod, is able to anchor a sphere. The two components self-assemble into a locally ordered monolayer consisting of an intercalated square lattice of octapods and spheres, which is reminiscent of the "tic-tac-toe" game. These tic-tac-toe domains form through an interfacial self-assembly that occurs by the dewetting of a hexane layer containing both particle types. By varying the experimental conditions and performing molecular dynamics simulations, we show that the ligands coating the octapods are crucial to the formation of this structure. We find that the tendency of an octapod to form an interlocking-type structure with a second octapod strongly depends on the ligand shell of the pods. Breaking this tendency by ligand exchange allows the octapods to assemble into a more relaxed configuration, which is able to form a lock-and-key-type structure with a sphere, when they have a suitable size ratio. Our findings provide an example of a more versatile use of branched nanocrystals in self-assembled functional materials.

Efficient algorithms for general periodic Lorentz gases in two and three dimensions

We present efficient algorithms to calculate trajectories for periodic Lorentz gases consisting of square lattices of circular obstacles in two dimensions, and simple cubic lattices of spheres in three dimensions; these become increasingly efficient as the radius of the obstacles tends to 0, the so-called Boltzmann–Grad limit. The 2D algorithm applies continued fractions to obtain the exact disc with which a particle will collide at each step, instead of using periodic boundary conditions as in the classical algorithm. The 3D version incorporates the 2D algorithm by projecting to the three coordinate planes. As an application, we calculate distributions of free path lengths close to the Boltzmann–Grad limit for certain Lorentz gases. We also show how the algorithms may be applied to deal with general crystal lattices.

Finite element simulation of arrays of hollow sphere structures

The numerical simulation of a repetitive model of a cellular metal is developed for fundamental investigation of the elastic behaviour of hollow sphere structures. Syntactic and partial morphologies of simple cubic configurations of hollow sphere structures are prescribed under boundary conditions corresponding to unit cells and periodic hollow sphere lattices. The obtained results are plotted as diagrams of Young's modulus versus relative density and allow to explain the behaviour of hollow sphere structures independent of their basic elastic properties.