Oblate Ellipsoids Research Articles

Collision rate of ice crystals with water droplets in turbulent flows

Riming, the process whereby ice crystals get coated by impacting supercooled liquid droplets, is one of the dominant processes leading to precipitation in mixed-phase clouds. How a settling crystal collides with very small water droplets has been mostly studied in laminar conditions. The present numerical study aims at providing further insight on how turbulent flow motion affects the riming of ice crystals. We model the crystals as narrow oblate ellipsoids, smaller than the Kolmogorov elementary scale. By neglecting the effect of fluid inertia on the motion of the crystals and droplets, and using direct numerical simulations of the Navier–Stokes equations in a moderately turbulent regime, over a range of kinetic energy dissipation$1~\text{cm}^{2}~\text{s}^{-3}\lesssim \unicode[STIX]{x1D700}\lesssim 256~\text{cm}^{2}~\text{s}^{-3}$, we determine the collision rate between disk-shaped ice crystals and very small liquid water droplets. Whereas differential settling plays the dominant role in determining the collision rate at small turbulence intensity, the role of turbulence becomes more important at the large values of$\unicode[STIX]{x1D700}$simulated, an effect that can be partly attributed to the increased role of inertia. We always find that collisions occur with a large probability on the rim of the ellipsoids, a phenomenon that can be explained to a large extent by kinematic considerations. The difference in the settling velocity of crystals and droplets induces a strong asymmetry in the probability of collision between the faces of the ellipsoids. Our results shed light on the physical mechanisms involved in the riming of ice crystals in clouds.

Finite Strain Analysis of the Wadi Fatima Shear Zone in Western Arabia, Saudi Arabia

Neoproterozoic rocks, Oligocene to Neogene sediments and Tertiary Red Sea rift-related volcanics (Harrat) are three dominant major groups exposed in the Jeddah tectonic terrane in Western Arabia. The basement complex comprises amphibolites, schists, and older and younger granites unconformably overlain by a post-amalgamation volcanosedimentary sequence (Fatima Group) exhibiting post-accretionary thrusting and thrust-related structures. The older granites and/or the amphibolites and schists display mylonitization and shearing in some outcrops, and the observed kinematic indicators indicate dextral monoclinic symmetry along the impressive Wadi Fatima Shear Zone. Finite strain analysis of the mylonitized lithologies is used to interpret the deformation history of the Wadi Fatima Shear Zone. The measured finite strain data demonstrate that the amphibolites, schists, and older granites are mildly to moderately deformed, where XZ (axial ratios in XZ direction) vary from 2.76 to 4.22 and from 2.04 to 3.90 for the Rf/φ and Fry method respectively. The shortening axes (Z) have subvertical attitude and are associated with subhorizontal foliation. The data show oblate strain ellipsoids in the different rocks in the studied area and indication bulk flattening strain. We assume that the different rock types have similar deformation behavior. In the deformed granite, the strain data are identical in magnitude with those obtained in the Fatima Group volcanosedimentary sequence. Finite strain accumulated without any significant volume change contemporaneously with syn-accretionary transpressive structures. It is concluded that a simple-shear deformation with constant-volume plane strain exists, where displacement is strictly parallel to the shear plane. Furthermore, the contacts between various lithological units in the Wadi Fatima Shear Zone were formed under brittle to semi-ductile deformation conditions.

Electromagnetic characteristics of systems of prolate and oblate ellipsoids

The present study suggests a novel model for simulating electromagnetic characteristics of spheroidal nanofillers. The electromagnetic interference shielding efficiency of prolate and oblate ellipsoids in the X-band frequency range is studied. Different multilayered nanocomposite configurations incorporating carbon nanotubes, graphene nanoplatelets, and carbon blacks are fabricated and tested. The best performance for a specific thickness is observed for the multilayered composite with a gradual increase in the thickness and electrical conductivity of layers. The simulation results based on the proposed model are shown to be in good agreement with the experimental data. The effect of filler alignment on shielding efficiency is also studied by using the nematic order parameter. The ability of a nanocomposite to shield the incident power is found to decrease by increasing alignment especially for high volume fractions of prolate fillers. The interaction of the electromagnetic wave and the fillers is mainly affected by the polarization of the electric field; when the electric field is perpendicular to the equatorial axis of a spheroid, the interaction is significantly reduced and results in a lower shielding efficiency. Apart from the filler alignment, size polydispersity is found to have a significant effect on reflected and transmitted powers. It is demonstrated that the nanofillers with a higher aspect ratio mainly contribute to the shielding performance. The results are of interest in both shielding structures and microwave absorbing materials.

The Interrelationship of Micro‐Meso and Macroscopic Structures on the Western Limb of the Hazara Kashmir Syntaxis, Pakistan

AbstractDetailed micro–meso to macroscopic structural analyses reveal two deformation phases in the western limb of the Hazara‐Kashmir Syntaxis (HKS). Bulk top to NW shearing transformed initially symmetrical NNE–SSW trending meso to macroscopic folds from asymmetric to overturned ones without changing their trend. Sigmoidal en‐echelon tension gashes developed during this deformation, that were oblique to bedding parallel worm burrows and bedding planes themselves. Strain analyses of deformed elliptical ooids using the Rf/ϕ method constrain the internal strain patterns of the NNE–SSW structures. The principal stretching axis (S3) defined by deformed elliptical ooids is oriented N27°E at right angles to WNW–ESE shortening. The deformed elliptical ooids in sub‐vertical bedding vertical planes contain ooids that plunge ∼70° SE due to NW‐directed tectonic transport. Finite strain ratios are 1.45 (Rxy) parallel to bedding plane and 1.46 (Ryz) for the vertical plane. From these 2D strain values, we derive an oblate strain ellipsoidal in 3D using the Flinn and Hsu/Nadai techniques. Strains calculated from deformed elliptical ooids average −18.10% parallel to bedding and −18.47% in the vertical plane. However, a balanced cross‐section through the study area indicates a minimum of ∼ −28% shortening. Consequently, regional shortening was only partially accommodated by internal deformation.

Ellipsoids (v1.0): 3-D magnetic modelling of ellipsoidal bodies

Abstract. A considerable amount of literature has been published on the magnetic modelling of uniformly magnetized ellipsoids since the second half of the nineteenth century. Ellipsoids have flexibility to represent a wide range of geometrical forms, are the only known bodies which can be uniformly magnetized in the presence of a uniform inducing field and are the only finite bodies for which the self-demagnetization can be treated analytically. This property makes ellipsoids particularly useful for modelling compact orebodies having high susceptibility. In this case, neglecting the self-demagnetization may strongly mislead the interpretation of these bodies by using magnetic methods. A number of previous studies consider that the self-demagnetization can be neglected for the case in which the geological body has an isotropic susceptibility lower than or equal to 0.1 SI. This limiting value, however, seems to be determined empirically and there has been no discussion about how this value was determined. In addition, the geoscientific community lacks an easy-to-use tool to simulate the magnetic field produced by uniformly magnetized ellipsoids. Here, we present an integrated review of the magnetic modelling of arbitrarily oriented triaxial, prolate and oblate ellipsoids. Our review includes ellipsoids with both induced and remanent magnetization, as well as with isotropic or anisotropic susceptibility. We also discuss the ambiguity between confocal ellipsoids with the same magnetic moment and propose a way of determining the isotropic susceptibility above which the self-demagnetization must be taken into consideration. Tests with synthetic data validate our approach. Finally, we provide a set of routines to model the magnetic field produced by ellipsoids. The routines are written in Python language as part of the Fatiando a Terra, which is an open-source library for modelling and inversion in geophysics.

Elastic properties of the nematic phase in hard ellipsoids of short aspect ratio.

We present a Monte Carlo simulation study of suspensions of hard ellipsoids of revolution. Based on the spatial fluctuations of the orientational order, we have computed the Frank elastic constants for prolate and oblate ellipsoids and compared them to the affine transformation model. The affine transformation model predicts the right order of magnitude of the twist and bend constant but not of the splay constant. In addition, we report the observation of a stable nematic phase at an aspect ratio as low as 2.5.

An ellipsoidal particle in tube Poiseuille flow

A suspended ellipsoidal particle inside a Poiseuille flow with Reynolds number up to 360 is studied numerically. The effects of tube diameter ($D$), inertia of the particle and the flow, and the particle geometry (both prolate and oblate ellipsoids) are considered. When a prolate particle with $a/b=2$ is inside a wider tube (e.g. $D/A>1.9$), where $A=2a$ is the length of the major axis of the particle, the terminal stable state is tumbling. When the prolate particle is inside a narrower tube ($1.0<D/A<1.9$), log-rolling or kayaking modes may appear. Which mode occurs depends on the competition between fluid and particle inertia. When the fluid inertia is dominant, the log-rolling mode appears, otherwise, the kayaking mode appears. Inclined and spiral modes may appear when $D/A<1$ and $D/A=1$, respectively. For a prolate ellipsoid with $a/b=4$, if $1<D/A<1.9$, there is only the kayaking mode and the log-rolling mode is not observed. When an oblate particle is inside a wider tube (e.g. $D/A>3.5$), it may adopt the log-rolling mode. Inclined and intermediate modes are firstly identified in narrower tubes. The phase diagram of the modes is also provided. The modes in the phase diagrams were not found to be affected by the initial state of the particle based on limited observation.

Conductivity of the 3D model of a composite with spheroidal inclusions

The solution to the problem of the conductivity of the 3D model of a composite with inclusion having a shape of oblate ellipsoids of revolution (spheroids) has been obtained in a wide range of concentrations. The entire range of variation of the shape of inclusions (from a sphere to an infinitely thin circular disk) has been considered. The relation between the percolation thresholds in this model and the quantities characterizing its effective conductivity in the limit of low concentrations of inclusions has been established.

A study to investigate pyrolysis of wood particles of various shapes and sizes

Pyrolysis of centimeter-scale wood particles of various sizes and shapes needs to be understood to determine their burning rate and life. Such particles may be thought of as firebrands, which are a major reason for spotting ignition in wildland and wildland-urban interface fires. The burning lifetime of firebrands controls the maximum distance they can travel to cause spotting. To understand and model this, experiments are done in a vertical tube furnace with wood particles of different sizes and shapes. For computations, two classes of shapes, prolate and oblate ellipsoids, were chosen to represent the arbitrary geometry of such particles. Prolate ellipsoids include shapes ranging from thin needles to spheres, whereas, oblate ellipsoids include shapes ranging from thin disks to spheres. The choice of these smooth shapes, while facilitating expedient computations also enables the coverage of wide ranges of particle shapes and surface area to volume ratios (SVR). Model simulations show satisfactory agreement with relevant literature and experimental data. Particle aspect ratio (ϵ, the ratio of minor and major axes), SVR, and equivalent radius (Re) are used to define the particle geometry. Mass loss and center temperature profiles are presented and discussed. It is shown that with the decreasing of aspect ratio, wood particle decomposes faster and the final char fraction becomes smaller. A power-law based correlation between conversion time (tcon) and SVR is derived and verified against experiments. Further, it is shown that an increase in the SVR enhances the production of tar and decreases the yield of char while leaving the yield of gas mostly unaffected.

Isotropic-nematic phase transition of uniaxial variable softness prolate and oblate ellipsoids.

Onsager's theory of the isotropic-nematic phase separation of rod shaped particles is generalized to include particle softness and attractions in the anisotropic interparticle force field. The procedure separates a scaled radial component from the angular integral part, the latter being treated in essentially the same way as in the original Onsager formulation. Building on previous treatments of more idealised hard-core particle models, this is a step toward representing more realistic rod-like systems and also allowing temperature (and in principle specific chemical factors) to be included at a coarse grained level in the theory. The focus of the study is on the coexisting concentrations and associated coexistence properties. Prolate and oblate ellipsoids are considered in both the small and very large aspect ratio limits. Approximations to the terms in the angular integrals derived assuming the very large (prolate) and very small (oblate) aspect ratios limits are compared with the formally exact treatment. The approximation for the second virial coefficient matches the exact solution for aspect ratios above about 20 for the prolate ellipsoids and less than ca. 0.05 for the oblate ellipsoids from the numerical evaluation of the angular integrals. The temperature dependence of the coexistence density could be used to help determine the interaction potential of two molecules. The method works at temperatures above a certain threshold temperature where the second virial coefficient is positive.

Highly anisotropic Cu oblate ellipsoids incorporated polymer composites with excellent performance for broadband electromagnetic interference shielding

In this study, highly anisotropic Cu oblate ellipsoids incorporated polymer composites were prepared that revealed excellent broadband electromagnetic shielding effectiveness of 80.0 to 62.1 dB at a frequency region between 300.0 KHz and 12.0 GHz, at low Cu contents. Cu-coated hollow polymer beads were fabricated through electroless plating of Cu on the polymer hollow beads. The hollow polymer beads were prepared through thermal expansion of acrylonitrile-based polymer beads containing a blowing agent. These beads were capable of incorporating highly anisotropic 2-dimensional (2D) Cu oblate ellipsoids into polymer composite through simple compression molding process. The resulting broadband electromagnetic shielding performance attributes to a low percolation behavior of the composites due to high anisotropy of the conductive Cu filler and their multilayered structure in the composites.

Morphological Evolution of Block Copolymer Particles: Effect of Solvent Evaporation Rate on Particle Shape and Morphology.

Shape and morphology of polymeric particles are of great importance in controlling their optical properties or self-assembly into unusual superstructures. Confinement of block copolymers (BCPs) in evaporative emulsions affords particles with diverse structures, including prolate ellipsoids, onion-like spheres, oblate ellipsoids, and others. Herein, we report that the evaporation rate of solvent from emulsions encapsulating symmetric polystyrene-b-polybutadiene (PS-b-PB) determines the shape and internal nanostructure of micron-sized BCP particles. A distinct morphological transition from the ellipsoids with striped lamellae to the onion-like spheres was observed with decreasing evaporation rate. Experiments and dissipative particle dynamics (DPD) simulations showed that the evaporation rate affected the organization of BCPs at the particle surface, which determined the final shape and internal nanostructure of the particles. Differences in the solvent diffusion rates in PS and PB at rapid evaporation rates induced alignment of both domains perpendicular to the particle surface, resulting in ellipsoids with axial lamellar stripes. Slower evaporation rates provided sufficient time for BCP organization into onion-like structures with PB as the outermost layer, owing to the preferential interaction of PB with the surroundings. BCP molecular weight was found to influence the critical evaporation rate corresponding to the morphological transition from ellipsoid to onion-like particles, as well as the ellipsoid aspect ratio. DPD simulations produced morphologies similar to those obtained from experiments and thus elucidated the mechanism and driving forces responsible for the evaporation-induced assembly of BCPs into particles with well-defined shapes and morphologies.

Calculation of debonding strength at the interface of particles with different shapes and matrix

The debonding process at the interface of filler-matrix is studied. The relevance of this process lays in its triggering effect of toughening mechanisms in polymer based composites. Based on a micromechanical model, a general approach of the debonding condition was derived. The proposed expression was applied for different particle shapes: spherical, ellipsoid, cubic, rectangular and fiber. Results of debonding strength versus geometric parameters of the different particles showed that oblate ellipsoids and cubic particles can be easily analyzed as spherical filler.

Impact of curvature on topological defects

We analyze the impact of extrinsic and intrinsic curvature on positions of topological defects (TDs) in two-dimensional (2D) nematic films. We demonstrate that both these curvature contributions are commonly present and are expected to be weighted by comparable elastic constants. A simple Landau-de Gennes approach in terms of tensor nematic order parameter is used to numerically demonstrate impact of the curvatures on position of TDs on 2D ellipsoidal nematic shells. In particular, in oblate ellipsoids the extrinsic and intrinsic elastic terms enforce conflicting tendencies to positions of TDs.

Acoustic radiation force on non-spherical objects in a liquid

Macro-scale acoustophoretic separation has applications in oil/water separation and biopharmaceutical cell clarification. Interesting phenomena include deformation of oil droplets and clustering of cells into cylindrical shapes. Particle and cluster size are not necessarily small relative to the wavelength. A general computational framework is needed to calculate the acoustic radiation force on particles, clusters, and cells of various shapes. Ilinskii et al. presented a method to evaluate an acoustic radiation force on spherical objects, based on expansion of incident and scattered field in spherical waves. The acoustic force is given by the incident wave amplitudes and dimensionless scattering coefficients. This method is generalized for particle shapes which are not spherical. The incident acoustic and scattered field are expanded with respect to spherical waves. An evaluation of the scattering amplitudes for non-spherical objects can be done numerically. Analytically, scattering amplitudes are calculated from boundary conditions for spheroidal functions that are solutions of a wave equation in spheroidal coordinates ξ, η, φ. Boundary conditions are defined at surface ξ= constant, i.e. on an ellipsoid surface. Solutions for prolate and oblate ellipsoids have been estimated. The asymptotic values of spheroidal solutions are determined for ξ large enough to obtain required amplitudes of scattered field. Ilinskii et al. POMA, 19 (2013).

Effect of magnetic nanoparticle shape on flux amplification in inductive coil magnetic resonance detection

We model and analyze the effect of particle shape on the signal amplification in inductive coil magnetic resonance detection using the reversible transverse magnetic susceptibility of oriented magnetic nanostructures. Utilizing the single magnetic domain Stoner-Wohlfarth model of uniform magnetization rotation, we reveal that different ellipsoidal particle shapes can have a pronounced effect on the magnetic flux enhancement in detection configurations typical of magnetic resonance settings. We compare and contrast the prolate ellipsoids, oblate ellipsoids, and exchange-biased spheres and show that the oblate ellipsoids and exchange-biased spheres have a significantly higher flux amplification effect than the prolate ellipsoids considered previously. In addition, oblate ellipsoids have a much broader polarizing magnetic field range over which their transverse flux amplification is significant. We show the dependence of transverse flux amplification on magnetic resonance bias field and discuss the resulting signal-to-noise ratio of inductive magnetic resonance detection due to the magnetic nanoparticle-filled core of the magnetic resonance detection coil.

Inclined transpression in the Neka Valley, eastern Alborz, Iran

Three major nappes in the Neka Valley in the eastern Alborz Mountains of Iran allow the Cimmerian to present convergence following the oblique collision between Iran and the southern margin of Eurasia. This work reports the identification of an inclined transpression zone recognized by field investigations and strain analyses of the geometries of formations and detailed mesoscopic structural analyses of multiple faults, folds and a cleavage. The main structures encountered include refolded recumbent asymmetric fold nappes, highly curved fold hinges, in a transpression zone that dips 37° to the NW between boundaries thrusts striking from N050° to N060°. The β angle (the angle between the zone boundary and direction of horizontal far-field shortening) is about 80°. The north-west and south-east boundaries of this zone coincide with the Haji-abad thrust and the Shah-Kuh thrust, respectively. Fold axes generally trend NE–SW and step to both right and left as a result of strike–slip components of fault displacements. Strain analyses using Fry’s method on macroscopic ooids and fusulina deformed into oblate ellipsoids indicate that the natural strain varies between 2.1 and 3.14. The estimated angle between the maximum instantaneous strain axis (ISAmax) and the transpression zone boundary (θ′) is between 6° and 20°. The estimated oblique convergence angle (α), therefore, ranges between 31° and 43°. The average kinematic vorticity number (W k ) is 0.6, in a zone of sinistral pure shear-dominated inclined triclinic transpression. These results support the applicability of kinematic models of triclinic transpression to natural brittle–ductile shear zones.

Multi-scale three-dimensional characterization of iron particles in dusty olivine: Implications for paleomagnetism of chondritic meteorites

Dusty olivine (olivine containing multiple sub-micrometer inclusions of metallic iron) in chondritic meteorites is considered an ideal carrier of paleomagnetic remanence, capable of maintaining a faithful record of pre-accretionary magnetization acquired during chondrule formation. Here we show how the magnetic architecture of a single dusty olivine grain from the Semarkona LL3.0 ordinary chondrite meteorite can be fully characterized in three dimensions, using a combination of focused ion beam nanotomography (FIB-nT), electron tomography, and finite-element micromagnetic modeling. We present a three-dimensional (3D) volume reconstruction of a dusty olivine grain, obtained by selective milling through a region of interest in a series of sequential 20 nm slices, which are then imaged using scanning electron microscopy. The data provide a quantitative description of the iron particle ensemble, including the distribution of particle sizes, shapes, interparticle spacings and orientations. Iron particles are predominantly oblate ellipsoids with average radii 242 ± 94 × 199 ± 80 × 123 ± 58 nm. Using analytical TEM we observe that the particles nucleate on sub-grain boundaries and are loosely arranged in a series of sheets parallel to (001) of the olivine host. This is in agreement with the orientation data collected using the FIB-nT and highlights how the underlying texture of the dusty olivine is crystallographically constrained by the olivine host. The shortest dimension of the particles is oriented normal to the sheets and their longest dimension is preferentially aligned within the sheets. Individual particle geometries are converted to a finite-element mesh and used to perform micromagnetic simulations. The majority of particles adopt a single vortex state, with “bulk” spins that rotate around a central vortex core. We observed no particles that are in a true single domain state. The results of the micromagnetic simulations challenge some preconceived ideas about the remanence-carrying properties of vortex states. There is often not a simple predictive relationship between the major, intermediate, and minor axes of the particles and the remanence vector imparted in different fields. Although the orientation of the vortex core is determined largely by the ellipsoidal geometry (i.e., parallel to the major axis for prolate ellipsoids and parallel to the minor axis for oblate ellipsoids), the core and remanence vectors can sometimes lie at very large (tens of degrees) angles to the principal axes. The subtle details of the morphology can control the overall remanence state, leading in some cases to a dominant contribution from the bulk spins to the net remanence, with profound implications for predicting the anisotropy of the sample. The particles have very high switching fields (several hundred millitesla), demonstrating their high stability and suitability for paleointensity studies.

Effect of filler alignment on percolation in polymer nanocomposites using tunneling-percolation model

In this study, we examine the effect of filler alignment on percolation behavior of polymer nanocomposites using Monte Carlo simulations of monodisperse prolate and oblate hard-core soft-shell ellipsoids representing carbon nanotubes and graphene nanoplatelets, respectively. The percolation threshold is observed to increase with increasing extent of alignment as expected. For a highly aligned system of rod-like fillers, the simulation results are shown to be in good agreement with the second virial approximation based predictions. However, for a highly aligned system of disk-like fillers, the second virial approximation based results are observed to significantly deviate from the simulations, even for higher aspect ratios. The effect of filler alignment on anisotropy in percolation behavior is also studied by predicting the percolation threshold along different directions. The anisotropy in percolation threshold is found to vanish even for highly aligned systems of fillers with increasing system size.

Influence of the exclusion distance among trees on gap fraction and foliage clumping index of forest plantations

A hypergeometric model is proposed explicitly instead of two previous stochastic models (the Poisson model and Neyman-A model) to describe the topological relationship of trees and the influence of the exclusion distance on gap fraction and clumping index of forest plantation canopies. Gap fraction (GF) and clumping index (CI) play key roles in plant light interception, and therefore they have strong impacts on plant growth and canopy radiative transfer processes. Trees are usually assumed to be randomly distributed in natural forests in many previous studies. However, few studies have shown how trees are distributed in forest plantations and how these distribution patterns affect GF and CI in these forests. In this paper, a simple and general distance factor defined as relative allowable shortest distance between centers of two adjacent crowns divided by the mean diameter of the crowns (RASD) is proposed to describe quantitatively the degree of mutual exclusion among trees in forest plantations of various tree distribution patterns. A hypergeometric model is proposed instead of two previous stochastic tree distribution models (the Poisson model and Neyman-A model) to describe the topological relationship of trees and the influences of the exclusion distance on the GF and CI of the forest plantation canopies. The results show that: (1) the hypergeometric model is more suitable than the Poisson model and Neyman-A model for describing the topological relationship of trees in forest plantations; (2) the exclusion distance has strong impacts on GF and CI: there are significant differences between the results of the hypergeometric model and the Poisson model. Larger RASD causes lower GF and larger CI. The simulations are verified by field measurements in four forest plantation stands. Similarly, impacts of RASD on GF and CI are also found for other two crown shapes (prolate and oblate ellipsoids).