Lattice Of Spheres Research Articles

Study of natural photonic crystals in beetle scales and their conversion into inorganic structures via a sol–gel bio-templating route

The origin of the structural colors from several different examples of the weevil and longhorn families (Curculionidae and Cerambycidae, respectively) was investigated by structural and optical characterization techniques. A range of interesting three-dimensional photonic crystal structures operating at visible wavelengths was discovered, including both disordered and ordered non-close-packed lattices of cuticular spheres and bicontinuous diamond-based architectures. The discovered photonic structures display a large variation in lattice constants and dielectric filling fractions and thereby create optical reflectance colors spanning the entire visible range. To transform these bio-polymeric photonic crystals into heat and photo-stable inorganic structures, a low-temperature bio-templating method was developed. Using organic–inorganic hybrid silica sol–gel infiltration–templation chemistry combined with acid-etching template removal, stable inverse photonic structures were fabricated. The inverse structures display good structural quality and vivid reflection properties.

Optical Analysis of the Fine Crystalline Structure of Artificial Opal Films

Herein, we present a detailed analysis of the structure of artificial opal films. We demonstrate that, rather than the generally assumed face centered cubic lattice of spheres, opal films are better approximated by rhombohedral assemblies of distorted colloids. Detailed analysis of the optical response in a very wide spectral range (0.4 < or = a/lambda < or = 2, where a is the conventional lattice constant), as well as at perpendicular and off-normal directions, unambiguously shows that the interparticle distance coincides very approximately with the expected diameter only along directions contained in the same close-packed plane but differs significantly in directions oblique to the [111] one. A full description of the real and reciprocal lattices of actual opal films is provided, as well as of the photonic band structure of the proposed arrangement. The implications of this distortion in the optical response of the lattice are discussed.

First-Principles Study of Casimir Repulsion in Metamaterials

We examine theoretically the Casimir effect between a metallic plate and several types of magnetic metamaterials in pursuit of Casimir repulsion, by employing a rigorous multiple-scattering theory for the Casimir effect. We first examine metamaterials in the form of two-dimensional lattices of inherently nonmagnetic spheres such as spheres made from materials possessing phonon-polariton and exciton-polariton resonances. Although such systems are magnetically active in infrared and optical regimes, the force between finite slabs of these materials and metallic slabs is plainly attractive since the effective electric permittivity is larger than the magnetic permeability for the studied spectrum. When lattices of magnetic spheres made from superparamagnetic composites are employed, we achieve not only Casimir repulsion but almost total suppression of the Casimir effect itself in the micrometer scale.

Inequalities for bounds on effective transport coefficients of two-phase media from power expansions at real points

The effective transport coefficient q of two-phase media is uniquely expressed by a special Stieltjes function f. In this paper, we incorporate into bounds on q, the power expansions of f available at a number of real points. To this end we apply the S-continued fraction technique developed in Baker (Essentials of Pade Approximants, Academic Press, New York, 1975) and Tokarzewski (IFTR Rep 4:3–171, 2005). Moreover, we establish the fundamental inequalities for S-bounds on q. As an example of an application the sequences of upper and lower S-bounds on the effective conductivity of a face-centered cubic lattice of spheres are evaluated.

CASIMIR ENERGY OF AdS5 ELECTROMAGNETISM AND COSMOLOGICAL CONSTANT PROBLEM

Casimir energy is calculated for the 5D electromagnetism in the warped geometry. It is compared with the flat case(arXiv:0801.3064). A new regularization, called sphere lattice regularization, is taken. It is based on the minimal area principle and is a direct realization of the geometrical approach to the renormalization group. The properly regularized form of Casimir energy, is expressed in a closed form. We numerically evaluate Λ(4D UV-cutoff), ω(5D bulk curvature, warp parameter) and T(extra space IR parameter) dependence of the Casimir energy. The warp parameter ω suffers from the renormalization effect. 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. We comment on the cosmological term at the end.

Electromagnetically induced transparency and slow light in an array of metallic nanoparticles

We present a classical analog of electromagnetically induced transparency occurring when light is absorbed by a two-dimensional lattice of metallic spheres mounted on an asymmetric dielectric waveguide. The transparency is manifested as a spectral hole within the surface-plasmon absorption peak of the metallic spheres and is a result of destructive interference of the waveguide modes with incident radiation. The presence of transparency windows is accompanied by slow light effect wherein the group velocity is reduced by a factor of 6000. At the same time, the minimum length for storing a bit of information is of the order of 100 nm. The proposed setup is a much easier means to achieve transparency and slow light compared to existing atomic, solid-state, and photonic systems and allows for the realization of ultracompact optical delay lines and buffers.

Casimir Energy of 5D Electromagnetism and New Regularization Based on Minimal-Area Principle

We examine the Casimir energy of 5D electromagnetism in the recent standpoint. The bulk geometry is flat. Z$_2$ symmetry and the periodic property, for the extra coordinate, are taken into account. After confirming the consistency with the past result, we do new things based on a {\it new regularization}. In the treatment of the divergences, we introduce IR and UV cut-offs and {\it restrict} the (4D momentum, extra coordinate)-integral region. The regularized configuration is the {\it sphere lattice}, in the 4D continuum space, which changes along the extra coordinate. The change (renormalization flow) is specified by the {\it minimal area principle}, hence this regularization configuration is string-like. We do the analysis not in the Kaluza-Klein expanded form but in a {\it closed} form. We do {\it not} use any perturbation. The formalism is based on the heat-kernel approach using the {\it position/momentum propagator}. Interesting relations between the heat-kernels and the P/M propagators are obtained, where we introduce the {\it generalized} P/M propagators. A useful expression of the Casimir energy, in terms of the P/M propagator, is obtained. The restricted-region approach is replaced by the weight-function approach in the latter-half description. Its meaning, in relation to the {\it space-time quantization}, is argued. {\it Finite} Casimir energy is numerically obtained. The compactification-size parameter (periodicity) suffers from the renormalization effect. Numerical evaluation is exploited. Especially the minimal surface lines in the 5D flat space are obtained both numerically using the Runge-Kutta method and analytically using the general solution.

Negative refraction of photonic and polaritonic waves in periodic structures

Negative refraction of photonic and polaritonic waves in periodic structuresNegative refraction can be achieved in photonic crystals. We briefly summarize recent studies in this field, and show that such effects are also possible in polaritonic and plasmonic structures, such as the dipole crystal. We propose a practical realization of this crystal, a periodic lattice of dielectric spheres. We study its mode structure, and preliminary results demonstrate the negative refraction on a polaritonic band.

A comparison of the GaAs atomic layer deposition infiltration of photonic crystals engineered by the controlled evaporation and Langmuir–Blodgett methods

Photonic crystal thin films were fabricated on glass substrates both by the controlled evaporation method and the Langmuir–Blodgett deposition of a lattice of silica spheres. Infilling of the air spaces within the structures with GaAs was achieved using trimethylgallium and arsine under atomic-layer-deposition conditions. The effect of infiltration on the (2 + 1) dimensional structure of Langmuir–Blodgett photonic crystals, as compared to the face-centred cubic structure of controlled evaporation photonic crystals, is directly investigated with respect to the observed optical properties.

CASIMIR ENERGY OF 5D ELECTRO-MAGNETISM AND SPHERE LATTICE REGULARIZATION

Casimir energy is calculated in the 5D warped system. It is compared with the flat one. The position/ momentum propagator is exploited. A new regularization, called sphere lattice regularization, is introduced. It is a direct realization of the geometrical interpretation of the renormalization group. The regularized configuration is closed-string like. We do not take the KK-expansion approach. Instead the P/M propagator is exploited, combined with the heat-kernel method. All expressions are closed-form (not KK-expanded form). Rigorous quantities are only treated (non-perturbative treatment). The properly regularized form of Casimir energy, is expressed in the closed form. We numerically evaluate its Λ(4D UV-cutoff), ω(5D bulk curvature, warpedness parameter) and T(extra space IR parameter) dependence.

Squeezing of a periodic emulsion through a cubic lattice of spheres

Squeezing of a periodic, highly concentrated emulsion of deformable drops through a dense, simple cubic array of solid spherical particles at zero Reynolds number is simulated by considering one drop in a periodic cell. The particles are rigidly held in space. The drops with nondeformed diameter comparable with the particle size (and considerably larger than the interparticle constrictions) squeeze under a specified average pressure gradient. This three dimensional problem serves as a useful prototype model of drop-solid interaction for emulsion flow through granular materials. The solution allows us to study permeabilities for both phases in detail and determine the critical conditions when the drop phase flow stops due to blockage in the pores by capillary forces. The algorithm employs a boundary-integral formulation with periodic Green’s function, Hebeker representation for solid-particle contributions, and recent desingularization tools [A. Z. Zinchenko and R. H. Davis, J. Fluid Mech. 564, 227 (2006)] to alleviate difficulties with lubrication. Calculations are challenging in that tens of thousands of boundary elements per surface and 10 000–20 000 time steps are required for near-critical squeezing conditions, and the use of multipole acceleration is crucial to make such simulations feasible. The results are presented for 36% and 50% concentrated emulsions flowing through an array of almost packed particles, at drop-to-medium viscosity ratios of 1 and 4. Scaling for the squeezing time of the drop phase at near-criticial capillary numbers is extracted from the calculations. For all the simulated cases, the drops move, on average, faster than the continuous phase.

Lyotropic Phase Behavior of Polybutadiene−Poly(ethylene oxide) Diblock Copolymers in Ionic Liquids

The lyotropic phase behavior of three poly(1,2-butadiene-b-ethylene oxide) diblock copolymers (PB−PEO) with different monomer volume fractions has been studied in two different ionic liquids, 1-ethyl-3-methylimidazolium bis(trifluoromethylsulfonyl)imide ([EMI][TFSI]) and 1-butyl-3-methylimidazolium hexafluorophosphate ([BMI][PF6]), across the complete concentration range. The ordered microstructures present in the solutions were characterized via small-angle X-ray scattering (SAXS). The phase diagrams for the PB−PEO/ionic liquid solutions include regions corresponding to the classical copolymer microstructures: body-centered-cubic lattices of spheres, hexagonally ordered cylinders, and lamellae. Additionally, the phase diagrams also include wide regions of coexisting microstructures and regions apparently corresponding to a disordered network microstructure. The phase behavior of the PB−PEO copolymers in both ionic liquids was comparable to their previously reported aqueous solution behavior. The temperature dependence of the phase diagrams was very modest, indicative of a highly segregated system. The level of solvent selectivity was also investigated via cryogenic transmission electron microscopy (cryo-TEM) on dilute solutions. On the basis of the morphology of the dilute solution copolymer aggregate structures in the ionic liquid solvents, and on the structural length scales of the concentrated solutions, it was concluded that for PB−PEO [BMI][PF6] behaves as a more selective solvent than [EMI][TFSI].

Thermal conductivity of a cubic lattice of spheres with capillary bridges

There have been many attempts to develop expressions for thermal conductivity (λ) as a function of liquid content (θ) in granular media such as powders, gravel and soil. We present a new approach based upon a cubic array of solid spheres, with liquid capillary bridges surrounding the solid–solid contacts. Numerical results agree with analytical solutions for the endpoints. For small filling angles (low liquid contents, such that menisci do not coalesce) and for a fixed conductivity ratio α ≡ λsolid/λliquid = λliquid/λgas, we find a simple relationship λ (θ) = λ (θ = 0) + aθb, where the prefactor and exponent are simple functions of α. The power-law relationship also holds for compressed spheres, if the dry conductivity λ (θ = 0) is adjusted for the new porosity. While the results are not trivially applicable to real granular media, they provide a quantitative explanation for the observed phenomenon in soils of the rapid rise in conductivity caused by the addition of a small volume of liquid.

Effective Conductivity of a Composite in a Primitive Tetragonal Lattice of Highly Conducting Spheres in Resistive Thermal Contact With the Isolating Matrix

A state-of-the-art study and a physical and numerical 3D finite element study of anisotropic conduction through composites filled with isometric inclusions of different conductivity were performed by modeling the longitudinal conduction across a tetragonal lattice of spheres in imperfect contact with the surrounding matrix. In dimensionless variables, the effective conductivity E is expressible as a function of a geometrical parameter B, reflecting the relative thickness of the gap between spheres, the Kapitza resistance C of the contact inclusion/matrix, and the relative resistivity D of the filler. The computation of some 600 E values at some 25 levels of the factors B, C, and D allows one to find some features, such as the leading role of the factor whose value is the highest of three, the low effect of the interactions between factors, the imperfect equivalence of the effects of the three factors, the slow and linear E dependence on the second and third greatest factor, and finally, the asymptotically exact linear relationship between E and the logarithmated sum of factors, with a slope depending only slightly on the relative magnitudes of factors.

Anisotropy of light propagation in thin opal films

Thin opal films are prepared by crystallization in a moving meniscus, and their optical transmission spectra are recorded in polarized light and studied. It is shown that the anisotropy of light propagation in the films is unambiguously related to the photonic band structure of opal and depends on the angle of incidence, the orientation of the incidence plane with respect to the opal lattice, and the wavelength and polarization of the incident light. Azimuthal diagrams of transmitted polarized light are constructed in the range of photonic band gaps of three orders for oblique incidence of a light beam. The anisotropy is found to vary with the light wave-length independently in perpendicular polarizations. A model of the band structure of opal wherein opal is represented as an fcc lattice of close-packed spheres adequately describes the optical transmission of opal films only in the range of the first-order photonic band gap.

Preparation, structural, and calorimetric characterization of bicomponent metallic photonic crystals

We report preparation and characterization of novel bicomponent metal-based photonic crystals having submicron three-dimensional (3D) periodicity. Fabricated photonic crystals include SiO2 sphere lattices infiltrated interstitially with metals, carbon inverse lattices filled with metal or metal alloy spheres, Sb inverse lattices, and Sb inverse lattices filled with Bi spheres. Starting from a face centered SiO2 lattice template, these materials were obtained by sequences of either templating and template extraction or templating, template extraction, and retemplating. Surprising high fidelity was obtained for all templating and template extraction steps. Scanning electron microscopy (SEM), small angle X-ray scattering (SAXS) and differential scanning calorimetry (DSC) were used to characterize the structure and the effects of the structure on calorimetric properties. To the best of our knowledge, SAXS data on metallic photonic crystals were collected for first time.

Photonic crystal thin films of GaAs prepared by atomic layer deposition

Photonic crystal thin films were fabricated via the self-assembly of a lattice of silica spheres on silicon (100) substrates. Progressive infilling of the air spaces within the structure with GaAs was achieved using trimethylgallium and arsine under atomic-layer-deposition conditions. Samples with the highest levels of GaAs infill were subsequently inverted using selective etching. Reflectance spectra are interpreted via the Bragg expression and calculated photonic band structure diagrams. For GaAs infilled and inverted samples, the relative positions of the first and second order Bragg reflections are strongly influenced by the wavelength dependent refractive index.

Photonic band gap in the face-centered cubic lattice of spherical shells connected by cylindrical tubes

We found a three-dimensional photonic crystal model, i.e., a non-close-packed face-centered cubic lattice of hollow spheres connected by thin cylindrical tubes, which possesses a complete three-dimensional photonic band gap between the eighth and ninth bands using photonic band structure calculations. The relative gap width reaches 9.1% at the low dielectric contrast of 7.84 appropriate to the transparent titania, and the minimum dielectric contrast needed to open the gap is about 5.3.

Segregation in Fluidized versus Tapped Packs

We compare the predictions of two different statistical mechanics approaches, corresponding to different physical measurements, proposed to describe binary granular mixtures subjected to some external driving (continuous shaking or tap dynamics). In particular we analytically solve at a mean field level the partition function of a simple hard sphere lattice model under gravity and focus on the phenomenon of size segregation. We find that the two approaches lead to similar results and seem to coincide in the limit of very low shaking amplitude. However, they give different predictions of the crossovers from Brazil nut effect to reverse Brazil nut effect with respect to the shaking amplitude, which could be detected experimentally.

Plasmon bands in metallic nanostructures

The photonic band structure of a three-dimensional lattice of metal spheres is calculated using an embedding technique, in the frequency range of the Mie plasmons. For a small filling factor of the spheres, Maxwell-Garnett theory gives an almost exact description of the dipole modes, and the multipole modes are fairly dispersionless. For a larger filling factor, crystal field effects modify the multipole frequencies, which show dispersion. These multipole bands are enclosed between the dipole modes. For touching spheres, there is a wide continuum of plasmon modes between zero frequency and the bulk metal plasmon frequency, which yield strong absorption of incident light. These plasmon modes are responsible for the blackness of colloidal silver.