Ellipsoids Of Revolution Research Articles

Scattering of nonspherical particles rebounding from a smooth and a rough surface in a high-speed gas-particle flow

Scattering indicatrices of nonspherical particles rebounding from a smooth and a rough surface are obtained by direct Monte Carlo simulations. Particles shaped as ellipsoids of revolution, rectangular prisms, and prisms with truncated vertices are considered. Surface roughness is defined as a two-dimensional profile whose scattering characteristics are close to those of real roughness induced by abrasive erosion of the surface in a high-speed gas-particle flow. Impact interaction of an individual particle with the surface is considered in a three-dimensional formulation. The scattering indicatrices of reflected particles are found to depend substantially on the particle shape in the case of rebound from a smooth surface and to be almost independent of the particle shape if the particles rebound from a rough surface.

A molecular dynamics study of chemical gelation in a patchy particle model.

We report event-driven molecular dynamics simulations of the irreversible gelation of hard ellipsoids of revolution containing several associating groups, characterizing how the cluster size distribution evolves as a function of the extent of reaction, both below and above the gel point. We find that over a very large interval of values of the extent of reaction, parameter-free mean-field predictions are extremely accurate, providing evidence that in this model the Ginzburg zone near the gel point, where non-mean field effects are important, is very limited. We also find that the Flory's hypothesis for the post-gelation regime properly describes the connectivity of the clusters even if the long-time limit of the extent of reaction does not reach the fully reacted state. This study shows that irreversibly aggregating asymmetric hard-core patchy particles may provide a close realization of the mean-field model, for which available theoretical predictions may help control the structure and the connectivity of the gel state. Besides chemical gels, the model is relevant to network-forming soft materials like systems with bioselective interactions, functionalized molecules and patchy colloids.

Effective Thermal Conductivity of Composite Materials with 3-D Microstructures and Interfacial Thermal Resistance

This article presents a numerical scheme, based on an isoparametric second-order finite-element discretization of the unit cell heat conduction problem, to calculate the effective thermal conductivity of composite materials with general 3-D microstructures and interfacial thermal resistance. Representative numerical results for the effective conductivity of ordered arrays of spheres, prolate ellipsoids of revolution, and finite-length circular cylinders are presented and, when possible, validated against previous analytical predictions. Furthermore, the effective conductivity of a disordered multiparticle cell is calculated, showing that the present computational approach is a valuable tool for understanding the macroscopic thermal behavior of composite materials.

Geometrical percolation of hard-core ellipsoids of revolution in the continuum

The percolation threshold of hard prolate ellipsoids of revolution dispersed in a continuum is obtained as a function of the aspect ratio. First random close packing of ellipsoids is produced by a dropping-and-shaking protocol. Two ellipsoids are regarded as connected when they come sufficiently close. Then a given fraction of ellipsoids selected randomly is removed and percolation of remaining ellipsoids is investigated as the fraction of remaining ellipsoids is varied. It is shown that the critical volume fraction of the colored ellipsoids is a decreasing function of the aspect ratio and that the aspect ratio dependence is well fitted by the inverse of the interaction range determined by the surface area and the radius of gyration of the ellipsoid surface.

Radiation-induced torques on spheroids

Radiation-induced torques on ellipsoids of revolution are discussed. Exact formulae for the thermal YORP torques are given in terms of elliptic integrals. It is demonstrated that in the absence of thermal inertia, the average values of dynamically significant projections of these torques are zero if a spheroid rotates around the axis of maximum inertia and if there are no resonances between rotation and orbital motion. The thermal lag leads to a systematic drift in the obliquity, but it does not affect the rotation period. The direct radiation pressure torques on spheroids are shown to be zero.

Mechanics of spheroidal fullerenes and carbon nanotubes for drug and gene delivery

There is considerable interest in the mechanics of carbon nanostructures, such as carbon nanotubes and fullerenes, and the manner of their interactions at the intermolecular level. Medical applications include the use of carbon nanotubes for targeted drug and gene delivery, for which issues relating to the acceptance and containment of drugs or genes are not properly understood. A spheroid is an ellipsoid with two equal axes and the general spheroidal shape includes a wide variety of possible molecular configurations such as spheres, capped cylindrical tubes and ellipsoids of revolution, and therefore the determination of the interaction forces for this general shape may have many applications. Phenomena such as the suction of fullerenes into carbon nanotubes due to the van der Waals interatomic interactions and ultra-low friction of a molecule moving inside a carbon nanotube give rise to the possibility of constructing nanoscaled oscillators with frequencies in the gigahertz range. This paper models the mechanics of such a system by employing a six-twelve Lennard-Jones potential taken over two surfaces assumed to be composed of mean distributions of atoms over the two idealized surfaces of an open-ended semi-infinite circular cylinder and a spheroid. Following the methodology of previous work with spherical surfaces, the acceptance energy and suction energy for spheroidal molecules are given and the special case of spherical molecules is also reproduced to validate the method. The results for elliptical molecules are novel and cannot be validated experimentally at this stage, but the results for the special case of spherical molecules are given and shown to be in good agreement with published molecular dynamical simulations. Finally, a general numerical-analytical procedure is proposed to calculate the Lennard-Jones potential for any axially symmetric surface, and the prior results obtained for the spheroid are used to validate the procedure.

Simple monoclinic crystal phase in suspensions of hard ellipsoids

We present a computer simulation study on the crystalline phases of hard ellipsoids of revolution. For aspect ratios >or=3 the previously suggested stretched-fcc phase [Frenkel and Mulder, Mol. Phys. 55, 1171 (1985)] is replaced by a different crystalline phase. Its unit cell contains two ellipsoids with unequal orientations. The lattice is simple monoclinic. The angle of inclination of the lattice, beta, is a very soft degree of freedom, while the two right angles are stiff. For one particular value of beta, the close-packed version of this crystal is a specimen of the family of superdense packings recently reported [Donev, Phys. Rev. Lett. 92, 255506 (2004)]. These results are relevant for studies of nucleation and glassy dynamics of colloidal suspensions of ellipsoids.

Study of gelatin–agar intermolecular aggregates in the supernatant of its coacervate

Intermolecular interaction leading to formation of aggregates between gelatin, a polyampholyte, and agar, a polysaccharide was studied in the supernatant of the complex coacervate formed by these biopolymers. Electrophoresis, laser light scattering and viscometry data were used to determine the interaction and the physical structure of these intermolecular soluble complexes by modeling these to be prolate ellipsoids of revolution (rod-like structures with well defined axial ratio and Perrin's factor). Solution ionic strength was found to reduce the axial ratio of these complexes implying the presence of screened polarization-induced electrostatic interaction between the two biopolymers.

Monitoring of the reflectors of ESA's Planck telescope by close-range photogrammetry

The Planck mission of the European Space Agency (ESA) is designed to image the anisotropies of the Cosmic Background Radiation Field over the whole sky. Planck's objective is to analyze, with the highest accuracy ever achieved, the remnants of the radiation that filled the universe immediately after the Big Bang, which we observe today as the cosmic microwave background. To achieve this aim well-manufactured reflectors are used as parts of the Planck telescope receiving system. The system consists of the Secondary and Primary Reflectors which are sections of two different ellipsoids of revolution with diameters of 1.1 and 1.9 meters. Deformations of the reflectors which influence the optical parameters and the gain of receiving signals are investigated in vacuum and at temperatures down to 95K, using close-range photogrammetric techniques. We have designed an optimal close-range photogrammetric network by heuristic simulation for the Primary and Secondary Reflectors with a mean relative precision better than 1:1,000,000 and 1:400,000, respectively, to achieve the requested accuracies. Special considerations have been taken into account in different steps of design, such as the determinability of additional parameters under the given network configuration, datum definition, reliability and precision issues as well as workspace limits and propagating errors from different sources of errors. A least squares best-fit ellipsoid was developed to determine the optical parameters of the reflector. We present our procedure and the results of processing the photogrammetric measurements of the Flight Models of the Primary and Secondary Reflectors which were executed by Thales Alenia Space France under ESA-ESTEC contract in vacuum and at very low temperatures.

The cut locus of a two-sphere of revolution and Toponogov's comparison theorem

We determine the structure of the cut locus of a class of two-spheres of revolution, which includes all ellipsoids of revolution. Furthermore, we show that a subclass of this class gives a new model surface for Toponogov's comparison theorem.

Simulation and Statistical Analysis of Random Packings of Ellipsoids

The paper describes an algorithm for the generation of random packings of ellipsoids of revolution, which is a natural generalization of an older algorithm for the case of spherical particles and belongs to the class of collective rearrangement algorithms. It yields packings with a broad spectrum of densities and geometrical properties, aspect ratio which depend on and the various parameters controlling the motions of ellipsoids during the simulation. The ellipsoid systems obtained are characterized by second-order characteristics, namely pair correlation and orientation correlation functions. Furthermore, the constructed packings are described in a phase diagram taken from statistical physics.

Void percolation and conduction of overlapping ellipsoids

The void percolation and conduction problems for equisized overlapping ellipsoids of revolution are investigated using the discretization method. The method is validated by comparing the estimated percolation threshold of spheres with the precise result found in literature. The technique is then extended to determine the threshold of void percolation as a function of the geometric aspect ratio of ellipsoidal particles. The finite element method is also applied to evaluate the equivalent conductivity of the void phase in the system. The results confirm that there are no universalities for void percolation threshold and conductivity in particulate systems, and these properties are clearly dependent on the geometrical shape of particles. As a consequence, void percolation and conduction associated with ellipsoidal particles of large aspect ratio should be treated differently from spheres.

Curvature of ellipsoids and other surfaces

From differential geometry one obtains an expression for the curvature in any direction at a point on a surface. The general theory is outlined. The theory is then specialised for surfaces that are represented parametrically as height over a transverse plane. The general ellipsoid is treated in detail as a special case. A quadratic equation gives the principal directions at the point and, hence, the principal curvatures associated with them. Equations are obtained for ellipsoids in general that are generalisations of Bennett's equations for sagittal and tangential curvature of ellipsoids of revolution. Equations are also presented for the locations of umbilic points on the ellipsoid.

Measuring the Three‐dimensional Structure of Galaxy Clusters. II. Are Clusters of Galaxies Oblate or Prolate?

The intrinsic shape of galaxy clusters can be obtained through a combination of X-ray and Sunyaev-Zeldovich effect observations once cosmological parameters are assumed to be known. In this paper we discuss the feasibility of modeling galaxy clusters as either prolate or oblate ellipsoids. We analyze the intracluster medium distribution for a sample of 25 X-ray-selected clusters, with measured Sunyaev-Zeldovich temperature decrements. A mixed population of prolate and oblate ellipsoids of revolution fits the data well, with prolate shapes preferred by ~60%-76%. We observe an excess of clusters nearly aligned along the line of sight, with respect to what is expected from a randomly oriented cluster population, which might imply the presence of a selection bias in our sample. We also find signs that a more general triaxial morphology might better describe the morphology of galaxy clusters. Additional constraints from gravitational lensing could disentangle the degeneracy between an ellipsoidal and a triaxial morphology, and could also allow an unbiased determination of the Hubble constant.

Intrinsic Viscosity of Proteins and Platonic Solids by Boundary Element Methods.

The boundary element (BE) method is used to implement a very precise computation of the intrinsic viscosity for rigid molecules of arbitrary shape. The formulation, included in our program BEST, is tested against the analytical Simha formula for ellipsoids of revolution, and the results are essentially numerically exact. Previously unavailable, very precise results for a series of Platonic solids are also presented. The formulation includes the optional determination of the center of viscosity; however, for globular proteins, the difference compared to the computation based on the centroid is insignificant. The main application is to a series of 30 proteins ranging in molecular weight from 12 to 465 kD. The computation starts from the crystal structure as obtained from the Protein Data Bank, and a hydration thickness of 1.1 Å obtained in previous work with BEST was used. The results (extrapolated to an infinite number of triangular boundary elements) for the proteins are separated into two groups: monomeric and multimeric proteins. The agreement with experimental measurements of the intrinsic viscosity in the case of monomeric proteins is excellent and within experimental error of 5%, demonstrating that the solution and crystal structure are hydrodynamically equivalent. However, for some multimeric proteins, we observe strong systematic deviations around -20%, which we interpret as a systematic deviation of the solution structure from the crystal structure. A possible description of the structural change is deduced by using simple ellipsoid model parameters. A method to obtain intrinsic viscosity values for proteins to 1-2% accuracy (better than experimental error) on the basis of a single BE computation (avoiding the need for an extrapolation on the number of surface triangles) is also presented.

Sequential quenching of randomly deposited ellipsoids: Anisotropy and spatial patterns

We studied the sequential quenching of prolate ellipsoids on a homogeneous surface by using our proposed pair potential whose repulsive part was that of hard ellipses and attractive part was the r −6 pairwise attraction. Both the strength and range parameters for the attraction were functions of the orientations of the pair of ellipses and related by ε ∝ σ − λ . The parameter λ determines the relative strength of the side-by-side and end-to-end attractions and thus plays an important role in determining the alignment of the particles. We adopted the value λ = 2.19 by using point-energy additivity to compute the minimum energies for both of these configurations for a pair of ellipsoids of revolution with aspect ratio of 2:1. We investigated the effect of temperature and the parameter λ on the alignments of ellipses. Both radial distribution function, g ( r ) , and orientational correlation function, G ( r ) , show the expected longer ranges of orientational correlation at lower temperatures and show higher degree of orientational order for λ = 3.5 than λ = 2.19 and 0.10. This can also be seen in the examples of configurations showing that for λ = 3.5 , ellipses are more aligned than λ = 2.19 and 0.1.

Demagnetization factors of the general ellipsoid: An alternative to the Maxwell approach

A transparent, exhaustive, and self-contained method for the calculation of the demagnetization tensor of the uniformly magnetized ellipsoid is presented. The method is an alternative to the established Maxwell derivation and is based on a Fourier-space approach to the micromagnetics of magnetized bodies. The key to the success of the procedure lies in the convenient treatment of shape effects through the Fourier representation. The scaled form of the demagnetization factors which depends on two dimensionless aspect ratios is argued to be their natural integral representation. Amongst other advantages, it allows for the immediate implementation of symmetry arguments such that only one of the principal factors needs to be computed. The oblate and prolate ellipsoids of revolution are examined from the same general point of view. The demagnetization factors for these distinct types of spheroid are seen to be related by analytic continuation of well-known Gaussian hypergeometric functions.

Decoupling Phenomena in Supercooled Liquids: Signatures in the Energy Landscape

A significant deviation from the Debye model of rotational diffusion in the dynamics of orientational degrees of freedom in an equimolar mixture of ellipsoids of revolution and spheres is found to begin at a temperature at which the average inherent structure energy of the system starts falling with drop in temperature. We argue that this onset temperature corresponds to the emergence of the process as a distinct mode of orientational relaxation. Further, we find that the coupling between rotational and translational diffusion breaks down at a still lower temperature where a change occurs in the temperature dependence of the average inherent structure energy.

Wave drag of an ellipsoid of revolution rapidly moving in a horizontal direction in a uniformly stratified fluid

The problem of the wave drag of ellipsoids moving in a uniformly stratified ideal fluid is considered by means of modeling the bodies by surface distributions of mass sources. Analytical results are obtained using the distributions known from the theory of a uniform fluid, which make it possible to describe the emission of internal waves by rapidly moving ellipsoids of revolution (spheroids) in the limit of large Froude numbers. An asymptotically simplified form of the dependence of the wave drag on the Froude number and the spheroid axis ratio is found. In the particular case of a sphere, the result confirmed earlier by Greenslade by making comparisons with a numerical calculation and experimental data is obtained.

Classification of dispersion equations for homogeneous, dielectric-magnetic, uniaxial materials

The geometric representation at a fixed frequency of the wave vector (or dispersion) surface omega(k) for lossless, homogeneous, dielectric-magnetic uniaxial materials is explored for the case when the elements of the relative permittivity and permeability tensors of the material can have any sign. Electromagnetic plane waves propagating inside the material can exhibit dispersion surfaces in the form of ellipsoids of revolution, hyperboloids of one sheet, or hyperboloids of two sheets. Furthermore, depending on the relative orientation of the optic axis, the intersections of these surfaces with fixed planes of propagation can be circles, ellipses, hyperbolas, or straight lines. The understanding obtained is used to study the reflection and refraction of electromagnetic plane waves due to a planar interface with an isotropic medium.