Ellipsoids Of Revolution Research Articles

Nematic-isotropic transition in a density-functional theory for hard spheroidal colloids.

We introduce a density-functional formalism based on the Parsons-Lee and the generalized van der Waals theories in order to describe the thermodynamics of anisotropic particle systems with steric interactions. For ellipsoids of revolution, the orientational distribution function is obtained by minimizing the free energy functional and the equations of state are determined. The system exhibits a nematic-isotropic discontinuous transition, characterized by a phase separation between nematic and isotropic phases at finite as well low packing fractions. The model presents a phase behavior which is in good agreement with Monte Carlo simulations for finite aspect ratios.

Scattering Properties of a Suspension Containing Plate-like Particles and Their Aggregates

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Laplace series for the level ellipsoid of revolution

The outer gravitational potential V of the level ellipsoid of revolution T is uniquely determined by two quantities: the eccentricity \(\varepsilon \) of the ellipsoid and Clairaut parameter q, proportional to the angular velocity of rotation squared and inversely proportional to the mean density of the ellipsoid. Quantities \(\varepsilon \) and q are independent, though they lie in a rather strict two-dimensional domain. It follows that Stokes coefficients \(I_n\) of Laplace series representing the outer potential of T are uniquely determined by \(\varepsilon \) and q. In this paper, we have found explicit expressions for Stokes coefficients via \(\varepsilon \) and q, as well as their asymptotics when \(n\rightarrow \infty \). If T does not coincide with a Maclaurin ellipsoid, then \(|I_n|\sim B\varepsilon ^n/n\) with a certain constant B. Let us compare this asymptotics with one of \(I_n\) for ellipsoids constrained by the only condition of increasing (even nonstrict) of oblateness from the centre to the periphery: \(|I_n|\sim \bar{B}\varepsilon ^n/(n^2)\). Hence, level ellipsoids with ellipsoidal equidensites do not exist. The only exception represents Maclaurin ellipsoids. It should be recalled that we confine ourselves by ellipsoids of revolution.

Microlocal analysis of imaging operators for effective common offset seismic reconstruction

The elliptic Radon transform (eRT) integrates functions over ellipses in 2D and ellipsoids of revolution in 3D. It thus serves as a model for linearized seismic imaging under the common offset scanning geometry where sources and receivers are offset by a constant vector. As an inversion formula of eRT is unknown we propose certain imaging operators (generalized backprojection operators) which allow to reconstruct some singularities of the searched-for reflectivity function from seismic measurements. We calculate and analyze the principal symbols of these imaging operators as pseudo-differential operators to understand how they map, emphasize or de-emphasize singularities. We use this information to develop local reconstruction operators that reconstruct relatively independently of depth and offset. Numerical examples illustrate the theoretical findings.

Numerical Simulation of the Percolation Threshold in Non-Overlapping Ellipsoid Composites: Toward Bottom-Up Approach for Carbon Based Electromagnetic Components Realization

A Monte Carlo (MC) model for the calculation of the percolation threshold in the composite filled with ellipsoids of revolution is developed to simulate the real experimental situation of percolative composites in which functional additives do not penetrate each other. The important advantage is that the MC model can be easily applied to multi-components composites, e.g., containing graphene nanoplatelets, carbon black and carbon nanotubes, by means of utilising the ellipsoids of different aspect ratios with the filling fraction corresponding to concentrations of each type of inclusion. The developed model could be used in a pre-experimental step for producing effective close-to percolation and percolated nanocomposites for various electromagnetic applications to avoid time and resources consuming the “sort-out” experimental phase of composition optimization, and could be utilized as the first step of the bottom-up material approach to touch the macroscopic platform for antennas/circuit realization.

An Ellipsoidal Model for Small Multilayer Particles

This paper presents an ellipsoidal model that is constructed for small layered nonspherical particles and methods for constructing “effective” multilayer ellipsoids, the light-scattering properties of which would be close to the corresponding properties of original particles. In the case of axisymmetric particles, prolate or oblate spheroids (ellipsoids of revolution) are implied. Numerical calculations of the polarizability and scattering cross sections of small layered nonspherical particles, including nonconfocal (similar) spheroids, Chebyshev particles, and pseudospheroids, are performed by different approximate and rigorous methods. Approximate approaches involve the use of an ellipsoidal model, in which the polarizability of a layered particle is determined in two ways. In the first case, the polarizability is calculated in the approximation of confocal spheroids, while, in the second case, it is sought as a linear combination of the polarizabilities of embedded spheroids proportionally to the volumes of layers. Among rigorous methods, the extended boundary conditions method and the generalized separation of variables method are applied. On the basis of a comparison of the results obtained with rigorous and approximate approaches, their drawbacks and advantages are discussed.

S.04.03 - Re-examining the antidepressant potential and mechanisms of isoflurane

The paper studies packing ellipsoids of revolution in a cylindrical container of minimum volume. Ellipsoids can be continuously rotated and translated. Two nonlinear mathematical programming models are introduced: exact and approximated. The latter uses an optimized multi-spherical approximation of ellisoids. For both models the phi-function technique is employed to describe analytically non-overlapping and containment constraints. Two solution approaches are proposed to solve the packing problem. Computational results for up to 500 ellipsoids are provided to demonstrate efficiency of the proposed approaches.

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.

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.

Characterization of the Pore Structure of Functionalized Calcium Carbonate Tablets by Terahertz Time-Domain Spectroscopy and X-Ray Computed Microtomography

Novel excipients are entering the market to enhance the bioavailability of drug particles by having a high porosity and, thus, providing a rapid liquid uptake and disintegration to accelerate subsequent drug dissolution. One example of such a novel excipient is functionalized calcium carbonate, which enables the manufacture of compacts with a bimodal pore size distribution consisting of larger interparticle and fine intraparticle pores. Five sets of functionalized calcium carbonate tablets with a target porosity of 45%-65% were prepared in 5% steps and characterized using terahertz time-domain spectroscopy and X-ray computed microtomography. Terahertz time-domain spectroscopy was used to derive the porosity using effective medium approximations, that is, the traditional and an anisotropic Bruggeman model. The anisotropic Bruggeman model yields the better correlation with the nominal porosity (R2 = 0.995) and it provided additional information about the shape and orientation of the pores within the powder compact. The spheroidal (ellipsoids of revolution) shaped pores have a preferred orientation perpendicular to the compaction direction causing an anisotropic behavior of the dielectric porous medium. The results from X-ray computed microtomography confirmed the nonspherical shape and the orientation of the pores, and it further revealed that the anisotropic behavior is mainly caused by the interparticle pores. The information from both techniques provides a detailed insight into the pore structure of pharmaceutical tablets. This is of great interest to study the impact of tablet microstructure on the disintegration and dissolution performance.

Laplace series of ellipsoidal figures of revolution

The theory of figures of equilibrium was extensively studied in the nineteenth century, when the reasons for which observed massive celestial bodies (such as the Sun, planets, and satellites) are almost ellipsoidal were discovered. The existence of exactly ellipsoidal figures was established. The gravitational potential of such figures can be represented by a Laplace series whose coefficients (the Stokes constants I n ) are determined by a certain integral operator. In the case of an ellipsoid of revolution with homothetic equidensites (surfaces of constant density), the general term of this series was found, and for some of the other mass distributions, the first few terms of the series were determined. In this paper, the general term of the series is found in the case where the equidensites are ellipsoids of revolution with oblateness increasing from the center to the surface. Simple estimates and asymptotics of the coefficients I n are also found. It turns out that the asymptotics depends only on the mean density, the density on the surface of the outer ellipsoid, and the oblateness of the outer ellipsoid.

A critical examination of the decoupling approximation for small-angle scattering from hard ellipsoids of revolution

The decoupling approximation, proposed by Kotlarchyk & Chen [J. Chem. Phys. (1983), 79, 2461–2469], is a first-order correction to the experimentally determined apparent structure factor that is necessary because of concentration effects in polydisperse and/or nonspherical systems. While the approximation is considered accurate for spheres with low polydispersity (<10%), the corresponding limitations for nonspherical particles are unknown. The validity of this approximation is studied for monodisperse dispersions of hard ellipsoids of revolution with aspect ratios ranging from 0.333 to 3 and a guide for its accuracy is provided.

Nondimensional maximum pressure gradient of tunnel compression waves generated by offset running axisymmetric trains

A high-speed train entering a tunnel generates a compression wave in the tunnel and a micro-pressure wave. The magnitude of the micro-pressure wave is approximately proportional to the maximum pressure gradient of the compression wave (∂p/∂t)max. High-speed railway operations require the consideration of the nose shape of the train in order to reduce (∂p/∂t)max generated by a train entering a tunnel. In this study, the compression waves generated by axisymmetric trains running at the offset position in double-track tunnels were investigated using a train launcher facility. Paraboloids of revolution, ellipsoids of revolution, and cones were used as the simplest nose shapes. The cone-nose train generated the largest value of (∂p/∂t)max, and the paraboloid-nose train generated the smallest value of (∂p/∂t)max among the three nose shapes. Although this tendency is the same as that for center running, the ratio of (∂p/∂t)max for the cone-nose train to that for the paraboloid-nose train became larger for offset running than center running. Moreover, the maximum value for the generation time of the compression wave was derived from acoustic theory, and the nondimensionalization of (∂p/∂t)max considering the nose length was proposed using it. Since the inverse of the nondimensionalized (∂p/∂t)max denotes the ratio of the generation time of the compression wave to its maximum value, it was defined as the efficiency of a train nose in the range of zero to unity. The values of the efficiency were almost constant with the nose length and 0.7, 0.6, and 0.5 for paraboloid-, ellipsoid-, and cone-nose shapes, respectively.

An Inverse Problem for the Refractive Surfaces with Parallel Lighting

In this article we examine the regularity of two types of weak solutions to a Monge--Ampere-type equation which emerges in a problem of finding surfaces that refract parallel light rays emitted from the source domain and striking a given target after refraction. Historically, ellipsoids and hyperboloids of revolution were the first surfaces to be considered in this context. The mathematical formulation commences with deriving the energy conservation equation for sufficiently smooth surfaces, regarded as graphs of functions to be sought, and then studying the existence and regularity of two classes of suitable weak solutions constructed from envelopes of hyperboloids or ellipsoids of revolution. Our main result in this article states that under suitable conditions on source and target domains and respective intensities these weak solutions are locally smooth.

Structure of Bolaamphiphile Sophorolipid Micelles Characterized with SAXS, SANS, and MD Simulations.

The micellar structure of sophorolipids, a glycolipid bolaamphiphile, is analyzed using a combination of small-angle X-ray scattering (SAXS), small-angle neutron scattering (SANS), and molecular dynamics (MD) simulations. Numerical modeling of SAXS curves shows that micellar morphology in the noncharged system (pH< 5) is made of prolate ellipsoids of revolution with core-shell morphology. Opposed to most surfactant systems, the hydrophilic shell has a nonhomogeneous distribution of matter: the shell thickness in the axial direction of the ellipsoid is found to be practically zero, while it measures about 12 Å at its cross-section, thus forming a "coffee bean"-like shape. The use of a contrast-matching SANS experiment shows that the hydrophobic component of sophorolipids is actually distributed in a narrow spheroidal region in the micellar core. These data seem to indicate a complex distribution of sophorolipids within the micelle, divided into at least three domains: a pure hydrophobic core, a hydrophilic shell, and a region of less defined composition in the axial direction of the ellipsoid. To account for these results, we make the hypothesis that sophorolipid molecules acquire various configurations within the micelle including bent and linear, crossing the micellar core. These results are confirmed by MD simulations which do show the presence of multiple sophorolipid configurations when passing from spherical to ellipsoidal aggregates. Finally, we also used Rb(+) and Sr(2+) counterions in combination with anomalous SAXS experiments to probe the distribution of the COO(-) group of sophorolipids upon small pH increase (5 < pH < 7), where repulsive intermicellar interactions become important. The poor ASAXS signal shows that the COO(-) groups are rather diffused in the broad hydrophilic shell rather than at the outer micellar/water interface.

Explicit excluded volume of cylindrically symmetric convex bodies.

We represent explicitly the excluded volume V(e){B(1),B(2)} of two generic cylindrically symmetric, convex rigid bodies, B(1) and B(2), in terms of a family of shape functionals evaluated separately on B(1) and B(2). We show that V(e){B(1),B(2)} fails systematically to feature a dipolar component, thus making illusory the assignment of any shape dipole to a tapered body in this class. The method proposed here is applied to cones and validated by a shape-reconstruction algorithm. It is further applied to spheroids (ellipsoids of revolution), for which it shows how some analytic estimates already regarded as classics should indeed be emended.

On the micromechanical modelling of the effective diffusion coefficient of a polycrystalline material

This study focuses on calculation of the effective diffusion coefficient of a polycrystalline material accounting for the grain size and shapes. Polycrystal is modelled as a composite consisting of a matrix with high diffusivity (grain boundaries and triple junctions) and inhomogeneities with low diffusivity (bulk grains including crystal defects like dislocations). The segregation at the grain boundaries is accounted for. Typical micromechanical models are re-written for diffusivity assuming that the grains have the shape of ellipsoids of revolution (spheroids). The results are compared with (1) numerical results for hydrogen diffusion in an imaginary polycrystalline material and (2) experimental results for diffusion of hydrogen in nickel polycrystal available in the literature. The approach can be used for extraction of information on diffusivity along the grain boundaries.

Design of a mirror for generating a prescribed continuous illuminance distribution based on the supporting quadric method

A method for designing a reflective surface to generate a prescribed continuous illuminance distribution is proposed. The optical surface is represented as an envelope of a two-parameter set of ellipsoids of revolution. The efficiency of the proposed method is illustrated with a case of computing and simulation of several reflective surfaces that generate uniform illuminance distributions shaped as an ellipse, a rectangle and a triangle.

Director alignment relative to the temperature gradient in nematic liquid crystals studied by molecular dynamics simulation.

The director alignment relative to the temperature gradient in nematic liquid crystal model systems consisting of soft oblate or prolate ellipsoids of revolution has been studied by molecular dynamics simulation. The temperature gradient is maintained by thermostating different parts of the system at different temperatures by using a Gaussian thermostat. It is found that the director of the prolate ellipsoids aligns perpendicularly to the temperature gradient whereas the director of the oblate ellipsoids aligns parallel to this gradient. When the director is oriented in between the parallel and perpendicular orientations a torque is exerted forcing the director to the parallel or perpendicular orientation. Because of symmetry restrictions there is no linear dependence of the torque being a pseudovector on the temperature gradient being a polar vector in an axially symmetric system such as a nematic liquid crystal. The lowest possible order of this dependence is quadratic. Thus the torque is very weak when the temperature gradient is small, which may explain why this orientation phenomenon is hard to observe experimentally. In both cases the director attains the orientation that minimises the irreversible entropy production.

A Model for Combustion of Firebrands of Various Shapes

The lifetime of a firebrand before burning out controls the maximum distance a firebrand can travel to cause spotting. Thus, combustion of firebrands of various shapes and sizes and their burnout time during transport is studied. The analysis assumes “quasi-steady” burning. In the present context, “quasi-steady” means that the rate processes controlling the gas phase fuel consumption and energy release are much faster than the particle fuel depletion time or the gas phase transport times. The Reynolds number based on the overall particle dimension and velocity relative to the ambient flow is assumed to be small. The gas phase combustion processes are represented by the evolution of a mixture fraction variable. It is shown that the velocity field near the particle can be described by a potential flow whose functional form is determined by the mass conservation equation and that this flow satisfies the particle surface boundary conditions. Gas phase solutions are obtained for two-parameter family of firebrand shapes composed of oblate and prolate ellipsoids of revolution. Prolate ellipsoids range from a thin needle to a sphere and oblate ellipsoids range from a sphere to a thin disc. Thus, they cover all possible firebrand shapes. The ambient velocity field does not need to be aligned with the firebrand axis of symmetry, so that the composite velocity and mixture fraction fields are three-dimensional. While a variety of steady-state condensed phase models are compatible with this picture, results are first presented for an ablating solid describable by the Spalding B number. B-numbers representative of flaming combustion of wood firebrands and glowing combustion of remaining char are used. All quantities are calculated as a function of ellipsoidal aspect ratio, B number, and the Reynolds number. Surprisingly, it is found that the firebrand burnout time is shape independent. All possible shapes were considered by using oblate and prolate ellipsoids of different sizes and aspect ratios. The burnout time depends only on the firebrand mass under the assumptions used.