Lattice Of Spheres Research Articles

A fundamental study of the extent of meaningful application of Maxwell׳s and Wiener׳s equations to the permeability of binary composite materials. Part II: A useful explicit analytical approach

The Maxwell equation relates the permeability P of a dispersion of particles A (modeled as hard congruent non-interacting spheres) in a continuous matrix B, to the ratio of the permeabilities of the components α=PA/PB and the corresponding volume fractions vA, vB (=1−vA). Originally devised for use at vA→0, the validity of the said equation for practical purposes was shown, by analytical methods, to extend to congruent weakly interacting spheres packed in regular cubic lattices, in the low to medium vA range; while the equation itself extends further up to the correct upper limit P=PA at vA=1. Replacing a simple cubic (s.c.) lattice of spheres with an identical lattice of cubes, overcomes this limitation (because cubes can pack up to vA=1) but only at the expense of losing analytical mathematical tractability. However, an analytical limiting form of the s.c. lattice of cubes could be derived near the limit vA→1 and shown to agree with the Maxwell equation. Even so, a large gap of unexplored territory was obviously still left in the remaining medium to high vA range. The gap in missing data was successfully filled in Part I by the use of a numerical computation approach, which showed that the result obtained at vA→1 is valid for practical purposes throughout the medium to high vA range; thus confirming the applicability of the Maxwell equation to sc lattices of cubes (and correspondingly of the Wiener equation in the case of anisometric particles) in this vA region. The gap in analytical treatment is filled to a large extent, in the present companion paper, by the development of an approach combining two different analytical limiting forms of the s.c. lattice-of-cubes model (as well as comparable limiting forms applicable to anisometric particles treated by the Wiener equation). We show here that this approach (referred to in the main text as the BP II model) provides (to a high degree of approximation in the practically important region of 0≤α≤10) a satisfactory analytical theoretical basis for meaningful application of the Maxwell equation in the higher vA range, comparable with that afforded by the existing lattice-of-spheres analytical treatments in the lower vA region.

Infrared thermal emission from a two-dimensional array of plasmonic spheres

We explore in numerical simulation the degree to which the thermal emission in the mid-infrared (IR) range can be controlled using a lattice of plasmonic spheres. While it has been shown that a great deal of spectral control can be achieved using a single sphere, obviously no directional control exists. We propose gaining this additional directional control over the thermal emission by designing a 2D lattice of resonant spheres. In this paper, we demonstrate that directional control as well as additionally spectral control over the thermal emission is achieved with this method. We do this by simulating a 2D layer of spheres composed of SiO2 cores and a SiC shells with a polaritonic resonance in the mid-IR. We then analyze how changing the lattice geometry affects the thermal emission characteristics of the ensemble of spheres. We show that for closely spaced spheres, modest directional control is possible with the additional benefit of some spectral control. When the spheres are spaced farther apart, Bragg scattering allows for more directional control, however, the spectrum of the emission is mostly governed by the properties of a single sphere.

TH‐C‐12A‐06: Feasibility of a MLC‐Based Inversely Optimized Multi‐Field Grid Therapy Technique

Purpose:Grid therapy (GT), which generates highly spatially modulated dose distributions, can deliver single‐ or hypo‐fractionated radiotherapy for large tumors without causing significant toxicities. GT may be applied in combination with immunotherapy, in light of recent preclinical data of synergetic interaction between radiotherapy and immunotherapy. However, conventional GT uses only one field, which does not have the advantage of multi‐fields in 3D conformal‐RT or IMRT. We have proposed a novel MLC‐based, inverse‐planned multi‐field 3D GT technique. This study aims to test its deliverability and dosimetric accuracy.Methods:A lattice of small spheres was created as the boost volume within a large target. A simultaneous boost IMRT plan with 8‐Gy to the target and 20‐Gy to the boost volume was generated in the Eclipse treatment planning system (AAA v10) with a HD120 MLC. Nine beams were used, and the gantry and couch angles were selected so that the spheres were perfectly aligned in every beams eye view. The plan was mapped to a phantom with dose scaled. EBT3 films were calibrated and used to measure the delivered dose.Results:The IMRT plan generated a highly spatially modulated dose distribution in the target. D95%, D50%, D5% for the spheres and the targets in Gy were 18.5, 20.0, 21.4 and 7.9, 9.8, 16.1, respectively. D50% for a 1cm ring 1cm outside the target was 2.9‐Gy. Film dosimetry showed good agreement between calculated and delivered dose, with an overall gamma passing rate of 99.6% (3%/1mm). The point dose differences for different spheres varied from 1–6%.Conclusion:We have demonstrated the deliverability and dose calculation accuracy of the MLC‐based inversely optimized multi‐field GT technique, which achieved a brachytherapy‐like dose distribution. Single‐fraction high dose can be delivered to the spheres in a large target with minimal dose to the surrounding normal tissue.

A second-order accurate immersed boundary-lattice Boltzmann method for particle-laden flows

A new immersed boundary-lattice Boltzmann method (IB-LBM) is presented for fully resolved simulations of incompressible viscous flows laden with rigid particles. The immersed boundary method (IBM) recently developed by Breugem (2012) [19] is adopted in the present method, development including the retraction technique, the multi-direct forcing method and the direct account of the inertia of the fluid contained within the particles. The present IB-LBM is, however, formulated with further improvement with the implementation of the high-order Runge–Kutta schemes in the coupled fluid–particle interaction. The major challenge to implement high-order Runge–Kutta schemes in the LBM is that the flow information such as density and velocity cannot be directly obtained at a fractional time step from the LBM since the LBM only provides the flow information at an integer time step. This challenge can be, however, overcome as given in the present IB-LBM by extrapolating the flow field around particles from the known flow field at the previous integer time step. The newly calculated fluid–particle interactions from the previous fractional time steps of the current integer time step are also accounted for in the extrapolation. The IB-LBM with high-order Runge–Kutta schemes developed in this study is validated by several benchmark applications. It is demonstrated, for the first time, that the IB-LBM has the capacity to resolve the translational and rotational motion of particles with the second-order accuracy. The optimal retraction distances for spheres and tubes that help the method achieve the second-order accuracy are found to be around 0.30 and −0.47 times of the lattice spacing, respectively. Simulations of the Stokes flow through a simple cubic lattice of rotational spheres indicate that the lift force produced by the Magnus effect can be very significant in view of the magnitude of the drag force when the practical rotating speed of the spheres is encountered. This finding may lead to more comprehensive studies of the effect of the particle rotation on fluid–solid drag laws. It is also demonstrated that, when the third-order or the fourth-order Runge–Kutta scheme is used, the numerical stability of the present IB-LBM is better than that of all methods in the literature, including the previous IB-LBMs and also the methods with the combination of the IBM and the traditional incompressible Navier–Stokes solver.

A fundamental study of the extent of meaningful application of Maxwell's and Wiener's equations to the permeability of binary composite materials. Part I: A numerical computation approach

In studies of the permeability of composite materials, consisting of solid isometric particles A dispersed in a polymeric matrix B, Maxwell's classical equation, rigorously valid only for very low fractional volumes (vA) of spherical particles, is often used for data analysis up to much higher vA values. Theoretical justification for this practice can be provided, only up to vA≈0.5, by current analytical models based on cubic lattices of congruent spheres. Replacing spheres by cubes yields a model covering the full composition range vA=0−1, but lacking convenient general analytical tractability. Accordingly, to explore unrestrictedly the practical validity of the Maxwell equation, a simple cubic lattice-of-cubes model was combined with an appropriate numerical computation tool. The results establish satisfactory applicability of the Maxwell equation for isometric particles A, at all compositions and over a wide range of component permeability ratios (α=PA/PB=0−100).The practical applicability of the corresponding classical equation of Wiener (which extends Maxwell's treatment to anisometric particles via the value of a single geometrical parameter AW) was similarly explored, using appropriate model s.c. lattices of (a) unidimensionally anisometric particles (transverse square rods of varying length) or (b) bidimensionally anisometric particles (transverse square platelets of varying area). The computations covered the same vA and α ranges as above, as well as a range of aspect ratios in each of the cases (a) and (b). The results obtained enable determination of values of AW, independent of vA and α, linked directly to the aspect ratio of the embedded particles, and demonstrate very good practical applicability of the Wiener equation (with the proper AW value) under all conditions studied.

Phonons in two-dimensional colloidal crystals with bond-strength disorder

We study phonon modes in two-dimensional colloidal crystals composed of soft microgel particles with hard polystyrene particle dopants distributed randomly on the triangular lattice. This experimental approach produces close-packed lattices of spheres with random bond strength disorder, i.e., the effective springs coupling nearest neighbors are very stiff, very soft, or of intermediate stiffness. Particle tracking video microscopy and covariance matrix techniques are then employed to derive the phonon modes of the corresponding "shadow" crystals with bond strength disorder as a function of increasing dopant concentration. At low frequencies, hard and soft particles participate equally in the phonon modes, and the samples exhibit Debye-like density of states behavior characteristic of crystals. For mid- and high-frequency phonons, the relative participation of hard versus soft particles in each mode is found to vary systematically with dopant concentration. Additionally, a few localized modes, primarily associated with hard particle motions, are found at the highest frequencies.

Theoretical Study on Interferometric Illumination of Gold Colloid-Sphere Monolayers to Produce Complex Structures for Spectral Engineering

Theoretical study on the interferometric illumination of colloid-sphere monolayers (IICSM) is presented to produce complex plasmonic structures consisting of wavelength-scaled periodic arrays of nano-objects with arbitrary array symmetry and controllable nanoscaled substructure. The IICSM method is based on illumination of hexagonal colloidsphere monolayers by interference patterns synchronized with sphere arrays along arbitrary preselected crystallographic directions. This nanokaleidoscope method enables tuning four structure parameters independently: the symmetry and characteristic periodicity of the interference pattern might be varied by the number, wavelength, and angle of incidence of the interfering beams; the distance between the nano-objects is controllable by the relative orientation of the interference pattern with respect to the hexagonal lattice of colloid spheres; the size of individual nano-objects is determined by the colloid-spheres' diameter and by the light wavelength; and the substructure size-parameter sensitively depends on the polarization state and can be tuned with the nano-object size simultaneously. Finite element method is applied to demonstrate the capabilities of IICSM based on gold colloid spheres and the impact of the resulted complex plasmonic patterns on spectral properties of thin gold films. The possibility of realizing spectral engineering with predesigned rectangular arrays of hole doublets that may be produced uniquely by IICSM is shown.

Casimir Energy of the Universe and the Dark Energy Problem

We regard the Casimir energy of the universe as the main contribution to the cosmological constant. Using 5 dimensional models of the universe, the flat model and the warped one, we calculate Casimir energy. Introducing the new regularization, called sphere lattice regularization, we solve the divergence problem. The regularization utilizes the closed-string configuration. We consider 4 different approaches: 1) restriction of the integral region (Randall-Schwartz), 2) method of 1) using the minimal area surfaces, 3) introducing the weight function, 4) generalized path-integral. We claim the 5 dimensional field theories are quantized properly and all divergences are renormalized. At present, it is explicitly demonstrated in the numerical way, not in the analytical way. The renormalization-group function (β-function) is explicitly obtained. The renormalization-group flow of the cosmological constant is concretely obtained.

Electrically conductive polymeric photonic crystals

Electrically conductive polymeric 3D photonic crystals are prepared by the shear ordering of composites consisting of monodisperse core-shell polymer spheres and single-walled carbon nanotubes (SWNTs). Strong iridescent colour indicates that the highly ordered opaline structures are not disrupted by the presence of the conductive nanotube networks. Thermal annealing leads to a significant increase in the overall electrical conductivity of thin-film samples yielding DC conductivities of 10−4 S cm−1, with percolation thresholds of less than 0.4 wt% of SWNT. Such composites with open networks of carbon nanotubes held apart by lattices of hard spheres, give combined conductive properties and structural colour effects, within a tuneable viscoelastic medium, with many potential functional applications.

Diffraction of Light by a Two-Dimensional Lattice of Spheres

Two-dimensional arrays of particles are of great interest because of their very characteristic optical properties and numerous potential applications. Although a variety of theoretical approaches are available for the description of their properties, methods that are accurate and convenient for computational procedures are always sought. In this work, a new technique to study the diffraction of a monochromatic electromagnetic field by a two-dimensional lattice of spheres is presented. The method, based on Fourier series, can take into account an arbitrary number of terms in the multipole expansion of the field scattered by each sphere. This method has the advantage of leading to simple formulas that can be readily programmed and used as a powerful tool for nanostructure characterization.

Fabrication of Ordered or Disordered Macroporous Structures with Various Ceramic Materials from Metal Oxide Nanoparticles or Precursors

Two different schemes were adopted to fabricate ordered macroporous structures with face centered cubic lattice of air spheres. Monodisperse polymeric latex suspension, which was synthesized by emulsifier-free emulsion polymerization, was mixed with metal oxide ceramic nanoparticles, followed by evaporation-induced self-assembly of the mixed hetero-colloidal particles. After calcination, inverse opal was generated during burning out the organic nanospheres. Inverse opals made of silica or iron oxide were fabricated according to this procedure. Other approach, which utilizes ceramic precursors instead of nanoparticles was adopted successfully to prepare ordered macroporous structure of titania with skeleton structures as well as lithium niobate inverted structures. Similarly, two different schemes were utilized to obtain disordered macroporous structures with random arrays of macropores. Disordered macroporous structure made of indium tin oxide (ITO) was obtained by fabricating colloidal glass of polystyrene microspheres with low monodispersity and subsequent infiltration of the ITO nanoparticles followed by heat treatment at high temperature for burning out the organic microspheres. Similar random structure of titania was also fabricated by mixing polystyrene building block particles with titania nanoparticles having large particle size followed by the calcinations of the samples.

The lattice parameter of the 28Si spheres in the determination of the Avogadro constant

The Avogadro constant has been determined by atom counting in two enriched 28Si spheres. Atoms were counted by exploiting their ordered arrangement in the spheres and calculating the ratio between sphere and the unit-cell volumes. This paper describes how the values of the sphere lattice parameters and, consequently, their unit-cell volumes were obtained.

A hybrid layer-multiple-scattering/Fourier modal method for photonic structures based on lithographic and/or self-assembly techniques

We present a new scattering method which allows for the calculation of the optical properties of lithography- and self-assembly-based photonic nanostructures. The lithographic components are modelled by the Fourier modal method while the self-assembly-based ones by the layer-multiple-scattering method. The key property which allows the merging of both theoretical methods into a common programming code are the scattering (transmission/reflection) matrices for a finite photonic slab. In order to demonstrate the potential of this new, hybrid method we study light absorption by a two-dimensional lattice of metallic rods with square cross-section sitting atop a homogeneous substrate and a two-dimensional lattice of dielectric spheres.

Effective transport properties of the porous electrodes in solid oxide fuel cells

This article presents a numerical framework for the computation of the effective transport properties of solid oxide fuel cells (SOFCs) porous electrodes from three-dimensional (3D) constructions of the microstructure. Realistic models of the 3D microstructure of porous electrodes are first constructed from measured parameters such as porosity and particle size distribution. Then each phase in the model geometries is tessellated with a computational grid. Three different types of grids are considered: Cartesian, octree, and body-fitted/cut-cell with successive levels of surface refinement. Finally, a finite volume method is used to compute the effective transport properties in the three phases (pore, electron, and ion) of the electrode. To validate the numerical approach, results obtained with the finite volume method are compared to those calculated with a random walk simulation for the case of a body-centred cubic lattice of spheres. Then, the influence of the sample size is investigated for random geometries with monosized particle distributions. Finally, effective transport properties are calculated for model geometries with polydisperse particle size distributions similar to those observed in actual SOFC electrodes.

Analysis of Double-Negative (DNG) Bandwidth for a Metamaterial Composed of Magnetodielectric Spheres Embedded in a Matrix

Based on Mie scattering theory and effective medium theory, the percentage bandwidth in which permittivity and permeability of a simple cubic lattice of identical, nonmetallic mag neto-dielectric spheres are simultaneously negative (double-negative, or DNG) is analyzed. Relaxing the restriction of previous work, that k <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> a → 0 where k <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> is the wavenumber in the matrix and α is the particle radius, new analytical formulas are obtained for k <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> a <; π/10, from which the DNG bandwidth can be easily computed. As an example, a contour map of DNG bandwidth for a wide range of values of particle permittivity and permeability is plotted for particle volume fraction f = 0.3. From the analytical formulas, it is shown that the maximum achievable DNG band width for a simple cubic lattice of spheres with particle volume fraction f = 0.3 is 7.4%, when the particle permittivity and permeability are both approximately 14 times greater than those of the matrix.

Optical Absorption in Overcoats of Nanoparticle Arrays on a Metallic Substrate

Surface plasma oscillations in metallic particles as well as in thin metallic films have been studied extensively in the past decades. New features regarding surface plasma excitations are, however, constantly discovered, leading, for example, to surface-enhanced Raman scattering studies and enhanced optical transmission though metal films with nanohole arrays. In the present work, the role of a metallic substrate is examined in two cases, one involving an overcoat of dielectric nanoparticles and the other an overcoat of metallic nanoparticles. Theoretical results are obtained by modeling the nanoparticles as forming a two-dimensional, hexagonal lattice of spheres. The scattered electromagnetic field is then calculated using a variant of the Green function method. Comparison with experimental results is made for nanoparticles of tungsten oxide and tin oxide deposited on either gold or silver substrates, giving qualitative agreement on the extra absorption observed when the dielectric nanoparticles are added to the metallic surfaces. Such absorption would be attributed to the mirror image effects between the particles and the substrate. On the other hand, calculations of the optical properties of silver or gold nanoparticle arrays on a gold or a silver substrate demonstrate very interesting features in the spectral region from 400 to 1,000 nm. Interactions between the nanoparticle arrays surface plasmons and their images in the metallic substrate would be responsible for the red shift observed in the absorption resonance. Moreover, effects of particle size and ambient index of refraction are studied, showing a great potential for applications in biosensing with structures consisting of metallic nanoparticle arrays on metallic substrates.

Suppression of the critical angle of diffraction in thin-film colloidal photonic crystals

We present an experimental and numerical investigation of the polarization anisotropy of the zero-order backdiffracted light from three-dimensional thin-film photonic crystals assembled from colloidal spheres. In particular, we compare simulations of reflectance spectra from perfectly ordered fcc lattice of spheres with measured reflectance data from self-organized opal films and forced-assembled Langmuir-Blodgett crystal films. We identify cross-polarization couplings and interactions between photonic crystal eigenmodes as the major physical mechanisms for resonance depolarization effects. Based on this, we find that a necessary condition for the observation of critical angles of diffraction in three-dimensional lattices is that the orientation of the light's plane of incidence coincides with a high-symmetry plane of the crystal lattice. We further show that for currently achievable colloidal photonic crystals with moderate refractive index contrast, this resonance depolarization mechanisms together with the destructive influence of lattice disorder effectively renders the meaning of a critical angle of diffraction obsolete.

Casimir energy of the universe and new regularization of higher dimensional quantum field theories

Casimir energy is calculated for the 5D electromagnetism and 5D scalar theory in the warped geometry. It is compared with the flat case. A new regularization, called sphere lattice regularization, is taken. In the integration over the 5D space, we introduce two boundary curves (IR-surface and UV-surface) based on the minimal area principle. It is a direct realization of the geometrical approach to the renormalization group. The regularized configuration is closed-string like. We do not take the KK-expansion approach. Instead, the position/momentum propagator is exploited, combined with the heat-kernel method. All expressions are closed-form (not KK-expanded form). The generalized P/M propagators are introduced. We numerically evaluate Λ(4D UV-cutoff), ω(5D bulk curvature, warp parameter) and T(extra space IR parameter) dependence of the Casimir energy. We present two new ideas in order to define the 5D QFT: 1) the summation (integral) region over the 5D space is restricted by two minimal surfaces (IR-surface, UV-surface); or 2) we introduce a weight function and require the dominant contribution, in the summation, is given by the minimal surface. Based on these, 5D Casimir energy is finitely obtained after the proper renormalization procedure. The warp parameter ω suffers from the renormalization effect. The IR parameter T does not. We examine the meaning of the weight function and finally reach a new definition of the Casimir energy where the 4D momenta (or coordinates) are quantized with the extra coordinate as the Euclidean time (inverse temperature). We examine the cosmological constant problem and present an answer at the end. Dirac's large number naturally appears.

Simulations of wave propagation and disorder in 3D non-close-packed colloidal photonic crystals with low refractive index contrast

The plane-wave expansion method (PWEM), the multiple-scattering method (MSM) and the 3D finite-difference time-domain method (FDTD) are applied for simulations of propagation of electromagnetic waves through 3D colloidal photonic crystals. The system investigated is not a "usual" artificial opal with close-packed fcc lattice but a dilute bcc structure which occurs due to long-range repulsive interaction between electrically charged colloidal particles during the growth process. The basic optical properties of non-close-packed colloidal PhCs are explored by examining the band structure and reflection spectra for a bcc lattice of silica spheres in an aqueous medium. Finite size effects and correspondence between the Bragg model, band structure and reflection spectra are discussed. The effects of size, positional and missing-spheres disorder are investigated. In addition, by analyzing the results of experimental work we show that the fabricated structures have reduced plane-to-plane distance probably due to the effect of gravity during growth.

Shear alignment and realignment of sphere-forming and cylinder-forming block-copolymer thin films

In common with many other structured fluids, block copolymers can be effectively oriented by shear. This susceptibility to shear alignment has previously been shown to hold even in thin films, containing as few as two layers of spherical microdomains, or even a single layer of cylindrical microdomains. A phenomenological model has been proposed [M. W. Wu, R. A. Register, and P. M. Chaikin, Phys. Rev. E 74, 040801(R) (2006)] to describe the alignment of such block-copolymer films, yielding the microdomain lattice order parameter as a function of shearing temperature, stress, and time. Here we directly test the central idea of that model, that the grains which are most misaligned with the shear direction are selectively destroyed, to reform in a direction more closely aligned with the shear. Films are first shear aligned from a polygrain state into a monodomain orientation and are then subjected to a second shear, at a variable stress (sigma) and misorientation angle (deltatheta) relative to the monodomain director, allowing the effects of sigma and deltatheta to be independently and systematically probed. For both cylinder-forming and sphere-forming block copolymers, these experiments confirm the basic premise of the model, as the stress required for realignment increases monotonically as deltatheta becomes smaller. For a cylinder-forming block copolymer, we find that the characteristic stress sigma(c) required to realign cylinders from one monodomain orientation to another is indistinguishable from that required to generate a monodomain orientation from the polygrain state. By contrast, the hexagonal lattice of spheres requires a value of sigma(c) more than 3 times as high for reorientation than for generation of the initial monodomain orientation.