Isotropic Sphere Research Articles

Stability and dynamics of chiral nanoparticles in lateral optical binding induced by high-order Bessel beams

The generalized multi-particle Mie equation (GMM) and electromagnetic momentum (EM) theory are applied to investigate the stability and dynamics of chiral nanoparticles in lateral optical binding induced by a high-order Bessel beam (HOBB). Such non-diffracting light suppressed the influence of the axial intensity profile of the illuminating beams on the self-organization process which then depended critically upon the inter-particles interactions. The illuminating HOBB is described in terms of beam shape coefficients (BSCs) within the framework of generalized Lorenz–Mie theories (GLMT). Utilizing the addition theorem of the vector spherical wave functions (VSWFs), the interactive scattering coefficients are derived through the continuous boundary conditions on which the interaction of the chiral nanoparticles is considered. The observed lateral binding force (BF) dependence of the separation of optically bound particles on the incidence of HOBB is in agreement with earlier theoretical prediction when the chiral spheres degenerate into isotropic spheres. We discuss the influence of the different parameters of the incident Bessel beam and of the chiral body on lateral BF in detail. Linearly and circularly polarized incident Bessel beams are considered, and the corresponding lateral BFs are compared and analyzed. The polarizations of incident HOBB considerably influence the lateral BF of chiral nanoparticles. In binding chiral nanoparticles, the polarization of incident beams should be chosen in accordance with the chirality. This finding may provide a recipe to understand the light interaction with multiple chiral particles of arbitrary shapes with the aid of the analytical approach. It could be a promising avenue in controlling the optical micromanipulation on chiral structures self-arrangement.

Classical problems in the theory of elasticity and the quantum theory of angular momentum

We show that applying the methods of the quantum theory of angular momentum enables us to obtain frequency equations for vibrational modes of a uniform isotropic elastic sphere, cylindrical rod, and infinite plate and results in a natural classification of these modes. We discuss how these models can be applied to describe vibrations of metal nanoparticles and semiconductor nanocrystals.

Azimuthal Imaginary Poynting Momentum Density.

The momentum of light beams can possess azimuthal densities, circulating around the beam axis and inducing intriguing mechanical effects in local light-matter interaction. Belinfante's spin momentum loops in circularly polarized beams, while the canonical momentum spirals in helically phased beams. However, a similar behavior of their imaginary counterpart, the so-called imaginary Poynting momentum (IPM), has not yet emerged. The foremost purpose of the present work is to put forward the discovery of this IPM vortex. We show that a simple superposition of radially and azimuthally polarized beams can form an IPM of completely azimuthal density. Additionally, the azimuthal IPM density can exist with a donut beam-intensity distribution and even with a vanishing azimuthal component of all other momenta. This uncovers the existence of a new mechanical effect which broadens the area of optical micromanipulation by achieving optical rotation of isotropic spheres, in the absence of both spin and orbital angular momenta. Our findings enrich the local dynamic properties of electromagnetic fields, highlighting the rotational action of their IPM, and thus its mechanical effect on microparticles and nanoparticles.

Scattering of an Airy light sheet by a chiral sphere.

Scattering of a 1D Airy beam light sheet by a chiral sphere is numerically studied within Mie theory and the plane-wave spectrum method. To testify to the validity of our code and method, the results of scattering intensity of a chiral sphere by an Airy beam light sheet reducing to a homogeneous isotropic sphere are compared with those in existing literature, which shows that these results are in good agreement. Influences of different parameters on differential scattering cross sections in the far field are investigated in detail, including the chiral parameters, sphere radius, and beam position. It is found in the scattering intensity of an Airy beam by a chiral sphere that the chiral sphere, which is compared with a homogeneous isotropic sphere, can decrease the scattering intensity in a region, and the scattering intensity distribution is sensitive to the x0 position of the Airy beam.

Effects of polycrystalline birefringent grains on the morphology dependent resonance modes of a spherical resonator

Morphology dependent resonance (MDR) modes in a dielectric sphere are highly sensitive to its shape, size, and refractive index (RI). Many dielectric materials with potential applications are birefringent in nature. Synthesis techniques of resonators of such materials may lead to polycrystalline morphologies. Due to its fascinating applications, titanium dioxide (TiO2) has attracted attention as a morphology dependent resonator. However, its high RI is accompanied by high birefringence. The effect of the grains on resonance modes is the interest of this study. Polycrystalline TiO2 microspheres with different grain distributions are synthesized by annealing at 500 °C and 700 °C. MDR modes in the photoluminescence spectrum of single spheres are found to exhibit mode splitting and mode shifting with respect to MDR modes of an equivalent isotropic sphere. The MDR spectral features of the near-perfect spheres have been investigated for the perturbative effects due to (a) nonsphericity and surface roughness and (b) inhomogeneity in RI arising from polycrystallinity and annealing conditions. Mode splits are demonstrated to arise from the perturbative effects of the larger grains through explicit computations using discrete dipole approximation for a Voronoi tessellated cell structure representing a polycrystalline sphere. Mode shifts are demonstrated to arise from radial inhomogeneity of the refractive index using the Aden-Kerker theory on spheres with core-shell morphology. The effects of surface roughness are not found to be significant. The present work extends the scope of model-assisted investigations in understanding morphology dependent resonators and indicates the possibility of designing resonators with prescribed refractive index features.

Supramolecular Spheres Self-Assembled from Conical Dendrons Are Chiral.

Frank-Kasper phases and liquid quasicrystals self-organize from supramolecular spheres of dendrimers, block copolymers, surfactants and other self-assembling molecules. These spheres are expected to be achiral due to their isotropic shape. Nevertheless, supramolecular spheres from short helical stacks of crown-like dendrimers self-organize a Pm3̅ n cubic (Frank-Kasper A15) phase which exhibits chirality on the macroscopic scale. However, the chirality of classic isotropic supramolecular micellar-like spheres, generated from conical dendrons, is unknown. Here we report a library of second and third generation biphenylpropyl dendrons with chiral groups at their apex that produces single-handed chiral supramolecular spheres. Up to 480 conical dendrons self-assemble to form micellar-like spheres, with a molar mass of up to 1.1 × 106 g/mol, that self-organize into a Pm3̅ n phase with chirality detectable on the macroscopic scale. This demonstration of chirality in micellar-like spheres of a Frank-Kasper phase raises the fundamental question whether micellar-like spheres forming 3D phases generated from other soft matter such as block copolymers, surfactants, and other molecules are chiral.

Characterization and correlation of signal drift in diffusion weighted MRI

Diffusion weighted MRI (DWMRI) and the myriad of analysis approaches (from tensors to spherical harmonics and brain tractography to body multi-compartment models) depend on accurate quantification of the apparent diffusion coefficient (ADC). Signal drift during imaging (e.g., due to b0 drift associated with heating) can cause systematic non-linearities that manifest as ADC changes if not corrected. Herein, we present a case study on two phantoms on one scanner. Different scan protocols exhibit different degrees of drift during similar scans and may be sensitive to the order of scans within an exam. Vos et al. recently reviewed the effects of signal drift in DWMRI acquisitions and proposed a temporal model for correction. We propose a novel spatial-temporal model to correct for higher order aspects of the signal drift and derive a statistically robust variant. We evaluate the Vos model and propose a method using two phantoms that mimic the ADC of the relevant brain tissue (0.36–2.2 × 10–3 mm2/s) on a single 3 T scanner. The phantoms are (1) a spherical isotropic sphere consisting of a single concentration of polyvinylpyrrolidone (PVP) and (2) an ice-water phantom with 13 vials of varying PVP concentrations. To characterize the impact of interspersed minimally weighted volumes (“b0's”), image volumes with b-value equal to 0.1 s/mm2 are interspersed every 8, 16, 32, 48, and 96 diffusion weighted volumes in different trials. Signal drift is found to have spatially varying effects that are not accounted for with temporal-only models. The novel model captures drift more accurately (i.e., reduces the overall change per-voxel over the course of a scan) and results in more consistent ADC metrics.

Convection and cracking stability of spheres in general relativity

In the present paper we consider convection and cracking instabilities as well as their interplay. We develop a simple criterion to identify equations of state unstable to convection, and explore the influence of buoyancy on cracking (or overturning) for isotropic and anisotropic relativistic spheres. We show that a density profile rho (r), monotonous, decreasing and concave , i.e. rho ' < 0 and rho '' < 0, will be stable against convection, if the radial sound velocity monotonically decreases outward. We also studied the cracking instability scenarios and found that isotropic models can be unstable, when the reaction of the pressure gradient is neglected, i.e. delta mathcal {R}_p = 0; but if it is considered, the instabilities may vanish and this result is valid, for both isotropic and anisotropic matter distributions.

Likely Cavitation in Stochastic Elasticity

We revisit the classic problem of elastic cavitation within the framework of stochastic elasticity. For the deterministic elastic problem, involving homogeneous isotropic incompressible hyperelastic spheres under radially symmetric tension, there is a critical dead-load traction at which cavitation can occur for some materials. In addition to the well-known case of stable cavitation post-bifurcation at the critical dead load, we show the existence of unstable snap cavitation for some isotropic materials satisfying Baker-Ericksen inequalities. For the stochastic problem, we derive the probability distribution of the deformations after bifurcation. In this case, we find that, due to the probabilistic nature of the material parameters, there is always a competition between the stable and unstable states. Therefore, at a critical load, stable or unstable cavitation occurs with a given probability, and there is also a probability that the cavity may form under smaller or greater loads than the expected critical value. We refer to these phenomena as ‘likely cavitation’. Moreover, we provide examples of homogeneous isotropic incompressible materials exhibiting stable or unstable cavitation together with their stochastic equivalent.

Ellipsoid Packing Structures on Freeform Surfaces

AbstractDesigners always get good inspirations from fascinating geometric structures gifted by the nature. In the recent years, various computational design tools have been proposed to help generate cell packing structures on freeform surfaces, which consist of a packing of simple primitives, such as polygons, spheres, etc. In this work, we aim at computationally generating novel ellipsoid packing structures on freeform surfaces. We formulate the problem as a generalization of sphere packing structures in the sense that anisotropic ellipsoids are used instead of isotropic spheres to pack a given surface. This is done by defining an anisotropic metric based on local surface anisotropy encoded by principal curvatures and the corresponding directions. We propose an optimization framework that can optimize the shapes of individual ellipsoids and the spatial relation between neighboring ellipsoids to form a quality packing structure. A tailored anisotropic remeshing method is also employed to better initialize the optimization and ensure the quality of the result. Our framework is extensively evaluated by optimizing ellipsoid packing and generating appealing geometric structures on a variety of freeform surfaces.

Humblet's angular momentum decomposition applied to radiation torque on metallic spheres using the Hagen–Rubens approximation

When a circularly polarized electromagnetic wave is scattered by an isotropic sphere, Humblet's decomposition from 1943 is helpful for understanding why the axial projection of the spin-angular momentum gives incomplete information concerning the total radiated angular momentum. Humblet asserted that some field angular momentum is orbital angular momentum (OAM). This distinction, applied here to Mie scattering, gives insight into why classical electromagnetic radiation torque is proportional to absorbed power for isotropic spheres. It also reveals the direct measurability (from Stokes parameters) of spin contributions to the radiated angular momentum. This approach is applied to metallic spheres in circularly polarized illumination modeled with Lorenz–Mie theory using the Hagen–Rubens approximation (1903) in which the real and imaginary components of the sphere's refractive index are equal in magnitude. That approximation is often useful for terahertz and lower frequencies. It was helpful to compare a spin efficiency factor Qspin with the canonical scattering efficiency factor Qsca for varying sphere sizes at a fixed frequency. A small-size limiting case Qsca/Qspin = 2 is recovered associated with a common interpretation of electric dipole-related torque. For larger sizes, the OAM contribution increases giving Qsca/Qspin > 2. Van de Hulst's scattering calculations for metal spheres are also reproduced.Dedicated in memory of Seattle Pacific University Professor James H. Crichton (1937–1999).

Thermal conductivity and tortuosity of porous composites considering percolation of porous network: From spherical to polyhedral pores

Understanding the effect of percolation behavior of complex geometrical pores on the tortuosity and thermal conductivity of porous composites is very crucial to the design and optimization of porous composites. In this work, we adopt the continuum percolation theory to accurately determine the nonlinear thermal conductivity and tortuosity of porous composites composed of homogeneous solid matrix and three-dimensional pores of geometrical morphologies from the isotropic sphere to anisotropic polyhedra. Through extensive Monte Carlo simulations and the finite-size scaling analysis, the percolation threshold of spherical and polyhedral pores is obtained. Two continuum percolation-based models are respectively presented to derive the tortuosity and thermal conductivity of porous composites over the whole porosities range, including near the percolation threshold. Comparison with extensive experimental, numerical and theoretical results confirms that the present models are capable of accurately determining the percolation threshold and tortuosity of complex geometrical porous networks and the effective thermal conductivity of porous composites as conductor-superconductor and insulator-conductor media. Furthermore, we use the proposed models to probe the influences of pore shape and porosity on the tortuosity and thermal conductivity of porous composites. The results elucidate the intrinsic interplay of component, structure, and thermal conductivity of porous composites, which can provide sound guidance for porous composite design and evaluation.

An analytical solution for the stress distribution within a spherically isotropic hollow sphere under diametrical compression

An analytical solution for the stress distributions within a spherically isotropic hollow sphere under diametrical compression is derived. The deformation of the spherically isotropic hollow sphere is divided into axis-symmetric deformation and non-symmetric deformation. The analytical solution for the axis-symmetric deformation is obtained by following the displacement method, while that for non-symmetric deformation of the spherically isotropic hollow sphere is obtained by employing the displacement function method. When the isotropic limit is considered, the analytical solution for isotropic hollow spheres is recovered identically. Numerical results show that tensile stress concentrations are usually developed both at the inner surface and near r/R = 0.85 within a hollow sphere along the axis of loading. A small value of the anisotropy in Young’s modulus or Poisson’s ratio usually leads to a large value of the maximum tensile stress at the inner surface. While a smaller value of the anisotropy in Poisson’s ratio or a large value of the anisotropy in the shear modulus may lead to a large value of the maximum tensile stress near r/R = 0.85 for a thick and anisotropic hollow sphere. In addition, it is found that the anisotropy in the shear modulus has drastic influence on the tensile stress distributions within a thin hollow sphere, and a particular value of the anisotropy in the shear modulus may reduce greatly the tensile stress concentrations both at inner surface and near r/R = 0.85 of a thin hollow sphere. This result may provide us a very attractive method to optimize the elastic constants of anisotropic hollow spheres by synthesis process technique so as to improve the load capacity of a hollow sphere, and extend the fatigue life of composite material made of thin hollow spheres.

Formation of Anisotropic Liquid Crystalline Nanoparticles via Polymerization-Induced Hierarchical Self-Assembly.

Polymeric nanoparticles (NPs) containing liquid crystalline (LC) mesogens with tunable anisotropic morphologies have applications in various fields, but their preparation typically suffers from tedious and low-throughput approaches. Here we present an efficient route to the preparation of anisotropic morphologies of azobenzene-containing block copolymers (BCPs) at high solids content via a polymerization-induced hierarchical self-assembly in ethanol. Various anisotropic NPs, including cuboids, short belts, lamellae, and ellipsoidal vesicles, have been obtained in a remarkably broad range of BCP compositions. The NPs exhibit a smectic phase with ordered stripes when observed under TEM. This internal LC ordering plays a significant role on the formation of these intriguing anisotropic morphologies. Morphological transitions from anisotropic to isotropic spheres can be obtained upon UV illumination due to the photoresponsive properties of the azobenzene mesogens. This work significantly expands the scope of accessible morphologies in PISA and suggests that the under explored LC BCPs may have an impactful role in the PISA field.

Anisotropic generalization of well-known solutions describing relativistic self-gravitating fluid systems: an algorithm

We present an algorithm to generalize a plethora of well-known solutions to Einstein field equations describing spherically symmetric relativistic fluid spheres by relaxing the pressure isotropy condition on the system. By suitably fixing the model parameters in our formulation, we generate closed-form solutions which may be treated as an anisotropic generalization of a large class of solutions describing isotropic fluid spheres. From the resultant solutions, a particular solution is taken up to show its physical acceptability. Making use of the current estimate of mass and radius of a known pulsar, the effects of anisotropic stress on the gross physical behaviour of a relativistic compact star is also highlighted.

Cracking isotropic and anisotropic relativistic spheres

We explore the influence of density fluctuations on isotropic and anisotropic configurations, extending the concept of cracking for isotropic general relativistic fluid spheres. This concept, conceived to describe the behaviour of anisotropic matter distributions just after its departure from equilibrium, could provide some insight on potential instabilities and future evolution of relativistic fluids. We have refined the idea of cracking, considering local fluctuations—represented by any function of compact support defined in a closed interval—and their effect on the state variables and gradients through “barotropic” equations of state, P = P(ρ) and [Formula: see text]. Under this approach it is found that both isotropic and anisotropic models could exhibit cracking (or overturning), and that previous results for cracking (and overturning) instabilities on anisotropic matter configurations has to be revised.

Revisiting the Fundamental Analytical Solutions of Heat and Mass Transfer: The Kernel of Multirate and Multidimensional Diffusion

AbstractThere are two types of analytical solutions of temperature/concentration in and heat/mass transfer through boundaries of regularly shaped 1‐D, 2‐D, and 3‐D blocks. These infinite‐series solutions with either error functions or exponentials exhibit highly irregular but complementary convergence at different dimensionless times, . In this paper, approximate solutions were developed by combining the error‐function‐series solutions for early times and the exponential‐series solutions for late times and by using time partitioning at the switchover time, . The combined solutions contain either the leading term of both series for normal‐accuracy approximations (with less than 0.003 relative error) or the first two terms for high‐accuracy approximations (with less than 10−7 relative error) for 1‐D isotropic (spheres, cylinders, slabs) and 2‐D/3‐D rectangular blocks (squares, cubes, rectangles, and rectangular parallelepipeds). This rapid and uniform convergence for rectangular blocks was achieved by employing the same time partitioning with individual dimensionless times for different directions and the product of their combined 1‐D slab solutions. The switchover dimensionless time was determined to minimize the maximum approximation errors. Furthermore, the analytical solutions of first‐order heat/mass flux for 2‐D/3‐D rectangular blocks were derived for normal‐accuracy approximations. These flux equations contain the early‐time solution with a three‐term polynomial in and the late‐time solution with the limited‐term exponentials for rectangular blocks. The heat/mass flux equations and the combined temperature/concentration solutions form the ultimate kernel for fast simulations of multirate and multidimensional heat/mass transfer in porous/fractured media with millions of low‐permeability blocks of varying shapes and sizes.

Solution of a Spherically Symmetric Static Problem of General Relativity for an Elastic Solid Sphere

The paper is concerned with the spherically symmetric static problem of the General Relativity Theory (GRT). The classical interior solution of this problem found in 1916 by K. Schwarzschild for a fluid sphere is generalized for a linear elastic isotropic solid sphere. The GRT equations are supplemented with the equation for the stresses which is similar to the compatibility equation of the theory of elasticity and is derived using the principle of minimum complementary energy for an elastic solid. Numerical analysis of the obtained solution is undertaken.

Pressure-Induced Disordering in SnO2 Nanoparticles

The high-pressure behavior of SnO2 nanoparticles (∼2.8 nm) was studied up to approximately 20 GPa using Raman spectroscopy in a diamond anvil cell and ab initio simulations. Above ∼7 GPa, the disordering, initially located at the surface, was found to propagate to the core of nanoparticles, ultimately leading to amorphous-like spectra. This observation can be interpreted as a disordering of the oxygen sublattice sensitively probed by Raman spectroscopy in contrast to powder X-ray diffraction techniques. The low-frequency mode can be related to the nanoparticle vibration as an elastic isotropic sphere motion. The pressure-induced shift of this mode allows for the constraining of the mechanical properties data reported in the literature.

Sorption Kinetics of CH4 and CO2 Diffusion in Coal: Theoretical and Experimental Study

Experimental and theoretical analyses with empirical correlations were framed for diffusion of gas species CH4 and CO2 in coal samples from Jharia coal fields, India, considering the intrinsic pore parameters. Coefficient of diffusion (D) and diffusivity (Deff) for a single and binary component coal–gas system were estimated by adopting unipore gas kinetic models for gas flow on the integration of Fick’s law and Langmuir relation. The rigorous study was carried out in estimating crossover pressure, which is dominant in distinguishing the flow regime for two primary types of diffusion: Knudsen and molecular as well as the transition between two regimes. Investigation reveals that experimental values of coefficients of diffusion of CH4 and CO2 in random homogeneous isotropic sphere packing of coal samples are in good agreement with the results of theoretical calculations. For the pressure range investigated, variation of coefficient of diffusion was found to follow a dual nature with a stable trend at press...