Oblate Ellipsoids Of Revolution Research Articles

The evolution of a polydisperse ensemble of ellipsoidal particles in the form of prolate and oblate ellipsoids of revolution

The evolution of a polydisperse ensemble of prolate and oblate ellipsoidal crystals in a supercooled one-component melt is theoretically studied. The volume growth rates for prolate and oblate ellipsoids are analytically found and compared at the same melt supercooling. We show that prolate crystals evolve faster than the oblate ones and the difference between their growth rates increases with increasing the melt supercooling. Then taking these volume growth rates into account, we formulate the model describing the evolution of an ensemble of prolate/oblate ellipsoidal particles. The analytical solution to this integrodifferential model is found for two nucleation mechanisms in cases of prolate and oblate ellipsoids using the saddle-point method. Our solution demonstrates that an ensemble of prolate particles grows and removes the melt supercooling faster than an ensemble of oblate particles. As a result, the particle-volume distribution function for prolate crystals is shifted to larger crystal volumes than the same distribution for oblate crystals. Keeping this behavior in mind, we conclude that the shape of crystals plays a decisive role in the melt supercooling dynamics and their volume distribution.

Equationof state of hard lenses: A combined virial series and simulation approach.

We provide highly accurate equation-of-state data for the isotropic phase of hard lenses obtained by means of cluster Monte Carlo simulations. This data is analyzed using a virial approach considering coefficients up to the order eight and Carnahan-Starling type closure relations for the virial series. The comparison with previously investigated systems consisting of hard, oblate ellipsoids of revolution allows insights into the detailed influence of the particle geometry. We propose a generalized Carnahan-Starling approach as a heuristic equationof state for the isotropic phase of hard lenses that in first approximation shows the same dependence on the excess part of the excluded volume as identified for oblate, hard lenses of revolution.

Trajectory Estimation for Particles Observed in the Vicinity of (101955) Bennu

AbstractWe analyze the trajectories of 313 particles seen in the near‐Bennu environment between December 2018 and September 2019. Of these, 65% follow suborbital trajectories, 20% undergo more than one orbital revolution around the asteroid, and 15% directly escape on hyperbolic trajectories. The median lifetime of these particles is ∼6 hr. The trajectories are sensitive to Bennu's gravitational field, which allows us to reliably estimate the spherical harmonic coefficients through degree 8 and to resolve nonuniform mass distribution through degree 3. The particles are perturbed by solar radiation pressure, enabling effective area‐to‐mass ratios to be estimated. By assuming that particles are oblate ellipsoids of revolution, and incorporating photometric measurements, we find a median axis ratio of 0.27 and diameters for equivalent‐volume spheres ranging from 0.22–6.1 cm, with median 0.74 cm. Our size distribution agrees well with that predicted for fragmentation due to diurnal thermal cycling. Detailed models of known accelerations do not produce a match to the observed trajectories, so we also estimate empirical accelerations. These accelerations appear to be related to mismodeling of radiation pressure, but we cannot rule out contributions from mass loss. Most ejections take place at local solar times in the afternoon and evening (12:00–24:00), although they occur at any time of day. We independently identify ten ejection events, some of which have previously been reported. We document a case where a particle ricocheted off the surface, revealing a coefficient of restitution 0.57±0.01 and demonstrating that some apparent ejections are not related to surface processes.

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.

Crystallization of Deformable Spherical Colloids

We introduce and characterize a first-order model for a generic class of colloidal particles that have a preferred spherical shape but can undergo deformations while always maintaining hard-body interactions. The model consists of hard spheres that can continuously change shape at fixed volume into prolate or oblate ellipsoids of revolution, subject to an energetic penalty. The severity of this penalty is specified by a single parameter that determines the flexibility of the particles. The deformable hard spheres crystallize at higher packing fractions than rigid hard spheres, have a narrower solid-fluid coexistence region and can reach high densities by a second transition to an orientationally ordered crystal.

Dynamics of a nonmagnetic drop suspended in a magnetic fluid in a rotating magnetic field

The behavior of a nonmagnetic drop suspended in a magnetic fluid and subjected to the action of a rotating magnetic field is studied experimentally. The configurations of the drop in the form of prolate and oblate ellipsoids of revolution in the rotating field are investigated, and disintegration of the drop in the rotating field and the development of comb instability of its surface under certain conditions are observed.

The spatial structure of dendritic macromolecules

A low-resolutionab initioshape determination was performed from small-angle neutron and X-ray scattering (SANS and SAXS) curves from solutions of polycarbosilane dendrimers with the three-functional and the four-functional branching centre of the fourth, fifth, sixth, seventh and eighth generations. In all cases, anisometric dendrimer shapes were obtained. The overall shapes of the dendrimers with the three- and four-functional branching centres were oblate ellipsoids of revolution and triaxial ellipsoids, respectively. The restored bead models revealed a pronounced heterogeneity within the dendrimer structure. The density deficit was observed in the central part and close to the periphery of the dendrimers. The fraction of the overall volume of the dendrimers available for solvent penetration was about 0.2–0.3. These results may help in the design of new practical applications of dendrimer macromolecules.

Magnetic properties of regular arrays of non-spherical ferromagnetic nanogranules

Magnetic properties (ground state, magnetic phase diagram, phase transitions in a magnetic field) of two- and three-dimensional lattices of ferromagnetic granules with dipole intergranule interaction are studied. The basic attention is given to lattices of non-spherical granules (prolate and oblate ellipsoids of revolution) and their limiting shapes—rod-like and disk-like granules. The analysis shows that conclusions of the theory agree with results of a series of experiments.

Lattices of nonspherical ferromagnetic grains with magnetic dipole interaction: Theory and experimental examples

The magnetic properties (ground state, magnetic phase diagram, and phase transitions in a magnetic field) of two-and three-dimensional lattices of ferromagnetic grains with the intergrain dipole interaction are studied. The main attention is paid to the lattices formed by nonspherical grains (prolate and oblate ellipsoids of revolution) and their extreme forms (rodlike and disc-shaped grains). An analysis shows that the conclusions of the theory are in good agreement with the results of experiments.

Asymptotics of Kolmogorov Diameters for Some Classes of Harmonic Functions on Spheroids

Let ΓKD be the unit ball of the space of all bounded harmonic functions in a domain D in R3, considered as a compact subset of the Banach space C(K), where K is a compact subset of D. The old problem about the exact asymptotics for Kolmogorov diameters (widths) of this set,lndk(ΓKD)∼−τk1/2,k→∞, is solved positively in the case when K and D are closed and open confocal spheroids, respectively (i.e., prolate or oblate ellipsoids of revolution). Using some special asymptotic formulas for the associated Legendre functions Pmn(coshσ) as n→∞ and m/n→γ∈[0, 1] (considered earlier by the second author), we show that the constant τ is some averaged characteristic of the pair of spheroids, expressed by means of a certain function of the variable γ, which appears within those asymptotics. Unlike the corresponding problem for analytic functions, quite well investigated, the harmonic functions case has been studied, up to now, only in the case of concentric balls.

Molecular interaction of disk-shaped colloidal particles

On the basis of relations of Lifshitz macroscopic theory of molecular forces, valid in case of the flat body interaction, and Deryaguin transition as well, making it possible to take account of a curvature of the separation surfaces of phases, an expression for the attraction force of the same size clay particles has been obtained. The particles have a shape of very oblate ellipsoids of revolution and, in the process of motion, are arranged with respect to each other so that their axes remain parallel. Analysis of the relationships obtained shows that the normal component of the attraction force is a monotonically decreasing function of the distance between the particles. The tangential component of that force changes with a change of the distance unmonotonically: at first, it increases from zero (at coaxial arrangement of the particles) to some maximum and then decreases tending to zero in the limit.

Calculating Phase Diagrams of Polymer−Platelet Mixtures Using Density Functional Theory: Implications for Polymer/Clay Composites

We analyze the thermodynamics of disklike particles dispersed in a polymer matrix. Using the Somoza−Tarazona free energy functional for thin oblate ellipsoids of revolution in combination with the ...

Perturbation theory for discotic nematic liquid crystals of axially symmetric molecules: effect of dispersion interaction

We investigate the influence of dispersion interaction on a variety of thermodynamic properties of discotic nematic liquid crystals at the discotic nematic-isotropic transition. We report calculations for a hard oblate ellipsoidal system, superposed with an attractive interaction represented by dispersion interaction subjected to different external pressures ranging from 1 to 300 bar. We consider a model system (which simulates a discotic nematic liquid crystal) in which molecules are assumed to interact via a pair potential having both repulsive and attractive parts. The repulsion part is represented by a repulsion between hard oblate ellipsoids of revolution and is a short range, rapidly varying potential. The attractive potential, a function of centre of mass distance and relative orientation between two molecules, is represented by dispersion interaction. The properties of the reference system and first order perturbation term are evaluated using a decoupling approximation which decouples orientational from translational degrees of freedom. The inclusion of fourth and sixth rank orientational order parameters in the calculation slightly improves the result. The role of pressure on phase transition parameters has also been studied.

Elastic constants of discotic (nematic) liquid crystals: effect of packing

Using a previously developed theory of nematic liquid crystals (Singh, Y., and Singh, K., 1986, Phys. Rev. A, 33, 3481) we present the calculation of elastic constants of discoticnematic liquid crystals and study the variation of elastic constants with packing fraction. The expressions for elastic moduli associated with 'splay', 'twist' and 'bend' modes of deformations are written in terms of order parameters characterizing the nature and amount of ordering in the phase and structural parameters. Numerical calculations are done for a model system, the molecules of which are hard oblate ellipsoids of revolution. It is observed that elastic constants are very sensitive to packing density and become larger with increase of shape anisotropy.

Prediction of Hydrodynamic Parameters of Biopolymers from Small-Angle Scattering Data

For correlating a number of structural and hydrodynamic parameters, biopolymers were modeled as prolate or oblate ellipsoids of revolution or as spheres (whole-body approaches). Axial ratios were estimated by combining the radius of gyration with either the hydrated volume or the surface-to-volume ratio. From these axial ratios and other parameters (volume, molar mass, partial specific volume), various hydrodynamic quantities were derived: frictional ratios, sedimentation and diffusion coefficients, Simha factors and intrinsic viscosities. In addition, the radii of several kinds of equivalent spheres (related to experimental volume and experimental or calculated values of diffusion coefficients and intrinsic viscosities) were computed and compared with each other and with the corresponding radii of gyration. Finally, the validity of simple empirical equations relating radius of gyration and diffusion coefficient or viscosity radius, sedimentation coefficient and molar mass was tested. As examples, the structural and hydrodynamic properties of a variety of globular biopolymers (such as proteins, viruses and ribonucleic acids) of different molar mass and shape were considered. Far-reaching conformity between observed and predicted values provided by the different techniques and approaches was achieved; for certain cases (e.g. hollow or inhomogeneous particles), however, more sophisticated procedures had to be applied. Predictive approaches were also used for hydrodynamic modeling of ligand-induced shape changes of enzymes. A representative example was elaborated in detail, including a critical assessment of errors.

Fluorescence resonance energy transfer study of shape changes in membrane-bound bovine prothrombin and meizothrombin.

Prothrombin activation to thrombin is a key control reaction in blood coagulation. During the process, prothrombin is sequentially cleaved at two peptide bonds (Arg323-Ile and Arg274-Thr) by factor X(a) to generate meizothrombin and then thrombin. Phosphatidylserine (PS)-containing membranes from platelets are believed to facilitate this two-step process. Using fluorescence energy transfer (FRET), we determined the distances of closest approach between a specifically located C-terminal fluorescein of a double mutant bovine prothrombin (P(S528A, G581C)-FM) or meizothrombin (M(S528A, G581C)-FM) and phosphatidylethanolamine-N-rhodamine B (PE-Rh; 0-8.7 mol %) contained in membranes composed of PS (25 mol %) and phosphatidylcholine (66.3-75 mol %). Plots of the energy transfer efficiency as a function of membrane concentration, at six PE-Rh surface densities, were analyzed globally to obtain dissociation constants and binding stoichiometries as global parameters and saturating energy transfer efficiencies characteristic of each surface density. From the global analysis, the dissociation constants were estimated to be 0.32 +/- 0.10 and 0.28 +/- 0.12 microM with stoichiometries of 42 +/- 12 and 44 +/- 9 lipid/protein for prothrombin and meizothrombin, respectively. The distance of closest approach was obtained from the dependence of the saturating energy transfer efficiency on the acceptor (PE-Rh) surface density. With the assumptions of kappa2 = 2/3 and n = 1.4, the distances were 94 +/- 3 A for prothrombin and 114 +/- 2 A for meizothrombin. Since both prothrombin and meizothrombin behave in solution as oblate ellipsoids of revolution with a long axis of 120 A, our FRET measurements suggest that binding to PS-containing membranes induced tighter folding of the prothrombin molecule but not of the meizothrombin intermediate. This observation is consistent with our hypothesis that membrane binding plays an essential role in the sequential alignment of the bond Arg323-Ile in prothrombin and Arg274-Thr in meizothrombin with the active site of the membrane-bound prothrombinase in the two-step thrombin-generating process.

Small-angle neutron-scattering study of micellar structure and interparticle interactions in Triton X-100 solutions.

Extensive measurements of small-angle nuetron-scattering cross sections from the nonionic micellar solutions of Triton X-100 in ${\mathrm{D}}_{2}$O are reported. An analysis of data at ambient temperature shows that the micelles are oblate ellipsoids of revolution with an axial ratio around two even at a concentration as low as 1% Triton X-100, and that there are about 20 molecules of ${\mathrm{D}}_{2}$O per surfactant molecule in the hydrophilic region. This micellar model---without invoking any further growth---is shown to explain the data for a wide range of concentrations at ambient temperature. Information about intermicellar interactions is obtained by analyzing the data using Baxter's sticky hard-sphere model, which assumes a square-well attractive potential between the micelles. This model is found to provide a qualitative interpretation of the temperature dependence of the data. However, important differences are noted for low momentum transfers. Some drawbacks of the model for characterizing the entire set of scattering data are brought out and discussed.

Third virial coefficient and the hard convex body equation of state

A modified equation of state of hard convex body fluids is proposed in which the ratio of the third virial coefficients is employed as the second parameter (in addition to the parameter of non-sphericity α); the resulting equation of state yields the dependence of the compressibility factor on density for hard convex body fluids with a considerable variety of shapes within the error estimates of the simulation data. The third virial coefficients of hard convex bodies of one type are expressed in terms of parameters α and τ; the proposed expressions discern between convex bodies of different shapes with the same values of α (e.g. prolate and oblate ellipsoids of revolution) and enable accurate determination of the third virial coefficients in a broad range of non-sphericities.

Equations of state for non-spherical hard-particle fluids

A new type of approximant, previously proposed for the equation of state of the hard-sphere fluid, is applied to obtain equations of state of non-spherical hard-particle fluids. For prolate and oblate spherocylinders, prolate and oblate ellipsoids of revolution, homonuclear and heteronuclear diatomics, the known virial coefficients are fitted as polynomials in the shape factor α and then the approximants are constructed from the fittings. The equations of state thus obtained are in very good agreement with existing simulation data.

More accurate equations of state for non-spherical hard-particle fluids

Equations of state for several non-spherical hard-particle fluids (prolate and oblate spherocylinders, prolate and oblate ellipsoids of revolution, homonuclear and heteronuclear diatomics) are developed for several geometrical shapes. For each type of particle, the known virial coefficients are fitted as polynomials in the shape factor α and then approximants, of a new type previously proposed for the hard sphere fluid, are constructed from the fittings. The equations of state thus obtained are in good to excellent agreement with existing simulation data and constitute a great improvement on other equations of state currently used for these fluids.