Internal and near-surface electromagnetic fields for a spheroidal particle with arbitrary illumination

A theoretical procedure in which a spheroidal coordinate separation-of-variables solution is used is developed for the determination of the internal and the near-surface electromagnetic fields for an arbitrary monochromatic field that is incident upon a homogeneous spheroidal particle. Calculations are presented for both the prolate and the oblate geometries, demonstrating the effects of particle size, particle axis ratio, and the orientation and character (plane-wave and focused Gaussian beam) of the incident field on the resultant internal and near-surface electromagnetic-field distributions.     

Stress analysis of a penny-shaped crack locted between two oblate spheroidal cavities in an infinite solid

The problem of a penny-shaped crack located between two oblate spheroidal cavities in an infinite solid subjected to uniaxial loads is considered. Using transformations between harmonic functions in cylindrical coordinates and those in oblate spheroidal ones, the problem is reduced to non-homogeneous linear equations. The obtained equations are solved numerically and the stress intensity factors at the penny-shaped crack tip under the influence of the two oblate spheroidal cavities are shown graphically.     

Theoretical investigation on heat transfer to burning char of spheroidal particles

The combustion of biomass fuel provides efficient and inexpensive utilization of solid biomass. Biomass particles generally exhibit non-spherical shapes, which affect the heat and mass transfer and further the char surface reactions. The prolate and oblate spheroidal particles are considered presently. The energy balance is employed for the gas phase near the particle in spheroidal coordinate system. A theoretical study is conducted on the char combustion of the spheroidal particles under either static conditions or forced convection. The expressions for the Nusselt numbers of both prolate and oblate spheroids are obtained. The char combustion rates for the spheroidal particles are affected by the Nusselt number and the particle surface area.     

Torsion of a round shaft of variable diameter

Problems for a round shaft of variable diameter subjected to torsion are studied. By transforming the governing equation into oblate spheroidal coordinates, a general solution is obtained in terms of the associated Legendre functions of order 2. Two illustrative problems, one being a shaft with a hyperbolic notch, the other a cylindrical shaft containing a small oblate spheroidal cavity located on its central axis, are solved. On the basis of the ensuing stress concentration, the important connection between the deep notch and crack is exploited. Several previously known solutions can be recovered. Results are extended to the cases of composites and transversely isotropic materials.     

Internal, near-surface, and scattered electromagnetic fields for a layered spheroid with arbitrary illumination

A spheroidal coordinate separation-of-variables solution has been developed for the determination of internal, near-surface, and scattered electromagnetic fields of a layered spheroid (either prolate or oblate) with arbitrary monochromatic illumination (e.g., plane wave or focused Gaussian beam). Calculated results are presented for layered 2:1 axis ratio prolate and oblate spheroids with an equivalent sphere size parameter of 20.     

Rainbow scattering by a cylinder with a nearly elliptical cross section

We both theoretically and experimentally examine the behavior of the first- and the second-order rainbows produced by a normally illuminated glass rod, which has a nearly elliptical cross section, as it is rotated about its major axis. We decompose the measured rainbow angle, taken as a function of the rod's rotation angle, into a Fourier series and find that the rod's refractive index, average ellipticity, and deviation from ellipticity are encoded primarily in the m = 0, 2, 3 Fourier coefficients, respectively. We determine these parameters for our glass rod and, where possible, compare them with independent measurements. We find that the average ellipticity of the rod agrees well with direct measurements, but that the rod's diameter inferred from the spacing of the supernumeraries of the first-order rainbow is significantly larger than that obtained by direct measurement. We also determine the conditions under which the deviation of falling water droplets from an oblate spheroidal shape permits the first few supernumeraries of the second-order rainbow to be observed in a rain shower.     

Analysis of diffusional broadening of vesicular packets of catecholamines released from biological cells during exocytosis.

Secretion of catecholamines is observed as a series of current spikes when measured at the surface of a bovine adrenal medullary cell in culture with a carbon-fiber microelectrode operated in the amperometric mode. Prior work has shown that these spikes are due to detection of concentrated packets of catecholamines which are released from individual vesicles after their fusion with the cell membrane, a process known as exocytosis. The shape of the individual current spikes, detected with a 5-microns spacing between the hemispherical cell and the electrode, has been compared to the shape generated by a theoretical model. The model consists of an instantaneous point source of material on a surface which subsequently diffuses to a disk that consumes the emitted material. The pertinent diffusion conditions have been evaluated with finite difference and random walk digital simulations. The two methods give identical results when the point source is located on a plane. The more realistic simulation geometry, emission from a hemispherical surface, was evaluated with the random walk method. The simulations allow a set of criteria to be established to evaluate diffusion-controlled broadening of spike shape. The broad range of spike widths observed experimentally and their individual shapes measured with 5-microns cell-electrode spacing are consistent with diffusion from point sources randomly distributed across a hemispherical surface. The data can be described with the diffusion coefficient for catecholamines in free solution. The model enables evaluation of signal-to-noise losses and correction for diffusional losses which are dependent on electrode radius.(ABSTRACT TRUNCATED AT 250 WORDS)     

The analytic solution for hydrodynamic diffraction by submerged prolate spheroids in infinite water depth

This study investigates the linear hydrodynamic scattering problem by stationary prolate spheroidal bodies and aims at providing an analytic solution for the associated boundary value problem. It extends the work of the present author on the hydrodynamics of oblate spheroidal bodies following the same procedure. The structural model under consideration is a spheroid with its polar axis greater than its equatorial diameter, subjected to the action of monochromatic incident waves. The polar axis is assumed to be perpendicular to the free surface that leads to the axisymmetric case concept. The analytic solution is sought using the method of multipole expansions constructed by employing Thorne’s formulas (Multipole expansions in the theory of surface waves. Proc Cam Philos Soc 49:707–716, 1953) that describe the velocity potential at singular points within a fluid domain with free upper surface and infinite water depth. The final stage of the solution process is the application of the zero velocity condition on the wetted surface of the spheroid. Inevitably this task requires the transformation of the involved velocity potentials, originally expressed with respect to spherical and polar coordinates, into prolate spheroidal coordinates. To this end, the appropriate addition theorems are derived, which recast Thorne’s expressions into infinite series of associated Legendre functions.     

Forced convection during feedback approach curve measurements in scanning electrochemical microscopy: maximal displacement velocity with a microdisk

In scanning electrochemical microscopy (SECM), an approach curve performed in feedback mode involves the downward displacement of a microelectrode toward a substrate while applying a bias to detect dissolved electroactive species at a diffusion-limited rate. The resulting measured current is said to be at steady state. In order to reduce the required measurement time, the approach velocity can be increased. In this paper, we investigate experimentally and theoretically the combination of diffusion and convection processes related to a moving microdisk electrode during feedback approaches. Transient modeling and numerical simulations with moving boundaries are performed, and the results are compared to the experimental approach curves obtained in aqueous solution. The geometry and misalignment of the microelectrode influence the experimental approach curves recorded at high approach velocities. The effects are discussed through the decomposition of the current into transient diffusional, radial convectional, and axial convectional contributions. Finally a ready-to-use expression is provided to rapidly evaluate the maximal approach velocity for steady state measurements as a function of the microelectrode geometry and the physical properties of the media. This expression holds for the more restrictive case of negative feedback as well as other modes, such as SECM approach curves performed at substrates displaying first order kinetics.     

Application of Mie theory to assess structure of spheroidal scattering in backscattering geometries

Inverse light scattering analysis seeks to associate measured scattering properties with the most probable theoretical scattering distribution. Although Mie theory is a spherical scattering model, it has been used successfully for discerning the geometry of spheroidal scatterers. The goal of this study was an in-depth evaluation of the consequences of analyzing the structure of spheroidal geometries, which are relevant to cell and tissue studies in biology, by employing Mie-theory-based inverse light scattering analysis. As a basis for this study, the scattering from spheroidal geometries was modeled using T-matrix theory and used as test data. In a previous study, we used this technique to investigate the case of spheroidal scatterers aligned with the optical axis. In the present study, we look at a broader scope which includes the effects of aspect ratio, orientation, refractive index, and incident light polarization. Over this wide range of parameters, our results indicate that this method provides a good estimate of spheroidal structure.     

Internal and near-surface electromagnetic fields for an absorbing spheroidal particle with arbitrary illumination

A previously developed [Appl. Opt. 34, 5542 (1995)] theoretical procedure for the calculation of the internal and the near-surface electromagnetic fields for nonabsorbing spheroidal particles with arbitrary monochromatic illumination has been generalized to permit solutions for absorbing (i.e., complex relative index of refraction) spheroidal particles. Calculations are presented that demonstrate the general effects of absorption on the internal and near-surface electromagnetic-field distributions for the particular case of a plane wave that is incident upon a 2:1-axis-ratio oblate spheroidal particle.     

Optical trapping of spheroidal particles in Gaussian beams

The T matrix method is used to compute equilibrium positions and orientations for spheroidal particles trapped in Gaussian light beams. It is observed that there is a qualitative difference between the behavior of prolate and oblate ellipsoids in linearly polarized Gaussian beams; the former generally orient with the symmetry axis parallel to the beam except at very small particle sizes, while the latter orient with the symmetry axis perpendicular to the beam. In the presence of a circularly polarized beam, it is demonstrated that oblate ellipsoids will experience a torque about the beam axis. However, for a limited range of particle sizes, where the particle dimensions are comparable with the beam waist, the particles are predicted to rotate in a sense counter to the sense of rotation of the circular polarization. This unusual prediction is discussed in some detail.     

Classification of simple amalgams

Taking into account the type of phase diagram of metal-mercury, solubility, heat of dissolution process, activity and diffusion coefficient of a metal in mercury as well as the kinetics of electroreduction of metallic aquo-cation on a mercury electrode with an amalgam formation, a general classification of simple amalgams into four groups is proposed. On this basis some experimentally unknown amalgam properties may be predicted which have significant meaning in technical and chemical applications.     

Stokes flow past a swarm of porous approximately spheroidal particles with Kuwabara boundary condition

The solution of the problem of symmetrical creeping flow of an incompressible viscous fluid past a swarm of porous approximately spheroidal particles with Kuwabara boundary condition is investigated. The Brinkman equation for the flow inside the porous region and the Stokes equation for the outside region in their stream function formulations are used. As boundary conditions, continuity of velocity and surface stresses across the porous surface and Kuwabara boundary condition on the cell surface are employed. Explicit expressions are investigated for both inside and outside flow fields to the first order in a small parameter characterizing the deformation. As a particular case, the flow past a swarm of porous oblate spheroidal particles is considered and the drag force experienced by each porous oblate spheroid in a cell is evaluated. The dependence of the drag coefficient on permeability for a porous oblate spheroid in an unbounded medium and for a solid oblate spheroid in a cell on the solid volume fraction is discussed numerically an and graphically for various values of the deformation parameter. The earlier known results are then also deduced from the present analysis.     

Electromagnetic fields for a spheroidal particle with an arbitrary embedded source

A spheroidal coordinate separation-of-variables solution has been developed for the determination of the internal, near-surface, and scattered fields of a spheroid (either prolate or oblate) with an embedded source of arbitrary type, location, and orientation. Presented results for (1,0) and (1,1) electric multipoles embedded in 2:1 axis ratio prolate and oblate spheroids (equal volume sphere size parameter equal to 20) illustrate that the presence of the spheroid interface can have a profound effect on the corresponding far-field scattering pattern. The calculation procedure could be used, for example, to model the emission of inelastic scattered light (Raman, fluorescence, etc.) from biological particles of appreciably elongated (prolatelike) or appreciably flattened (oblatelike) geometries.     

Ice-crystal absorption: a comparison between theory and implications for remote sensing

The problem of the disagreement between cirrus crystal sizes determined remotely and by in situ measurements is shown to be due to inappropriate application of Mie theory. We retrieved the absorption optical depth at 8.3 and 11.1 mum from 11 tropical anvil cirrus clouds, using data from the High Resolution Infrared Radiation Sounder (HIRS). We related the absorption optical depth ratio between the two wavelengths to crystal size (the size was defined in terms of the crystal median mass dimension) by assuming Mie theory applied to ice spheres and anomalous diffraction theory (ADT) applied to hexagonal columns, hexagonal plates, bullet rosettes, and aggregates (polycrystals). The application of Mie theory to retrievals yielded crystal sizes approximately one third those obtained with ADT. The retrievals of crystal size by use of HIRS data are compared with measurements of habit and crystal size obtained from in situ measurements of tropical anvil cirrus particles. The results of the comparison show that ADT provides the more realistic retrieval. Moreover, we demonstrate that at infrared wavelengths retrieval of crystal size depends on assumed habit. The reason why Mie theory predicts smaller sizes than ADT is shown to result from particle geometry and enhanced absorption owing to the capture of photons from above the edge of the particle (tunneling). The contribution of particle geometry to absorption is three times greater than from tunneling, but this process enhances absorption by a further 35%. The complex angular momentum and T-matrix methods are used to show that the contribution to absorption by tunneling is diminished as the asphericity of spheroidal particles is increased. At an aspect ratio of 6 the contribution to the absorption that is due to tunneling is substantially reduced for oblate particles, whereas for prolate particles the tunneling contribution is reduced by 50% relative to the sphere.     

Constitutive models for ductile solids reinforced by rigid spheroidal inclusions

We study the effective constitutive response of composite materials made of rigid spheroidal inclusions dispersed in a ductile matrix phase. Given a general convex potential characterizing the plastic “in the context of J₂-deformation theory” behavior of the isotropic matrix, we derive expressions for the corresponding effective potentials of the rigidly reinforced composites, under general loading conditions. The derivation of the effective potentials for the nonlinear composites is based on a variational procedure developed recently by Ponte Castaneda (1991a, J. Mech. Phys. Solids 39, 45–71). We consider two classes of composites. In the first class, the spheroidal inclusions are aligned, resulting in overall transversely isotropic symmetry for the composite. In the second class, the inclusions are randomly oriented, and thus the composite is macroscopically isotropic. The effective response of composites with aligned inclusions depends on both the orientation of the loading relative to the inclusions and on the inclusion concentration and shape. Comparing the strengthening effects of rigid oblate and prolate spheroids, we find that prolate spheroids give rise to stiffer effective response under axisymmetric “relative to the axis of transverse isotropy” loading, while oblate spheroids provide greater reinforcement for materials loaded in transverse shear. On the other hand, nearly spherical “slightly prolaterd spheroids are most effective in strengthening the composite under longitudinal shear. Thus, the optimal shape for strengthening composites with aligned inclusions depends strongly on the loading mode. Alternatively, the properties of composites with randomly oriented spheroidal inclusions, being isotropic, depend only on the concentration and shape of the inclusions. We find that both oblate and prolate inclusions lead to significant strengthening for this class of composites.     

Resistivity Measurement of a Small-Volume Sample Using Two Planar Disc Electrodes and a New Geometric Factor

In this work, we derive a new geometric factor in oblate spheroidal coordinates for conductivity or resistivity measurements using two planar disc electrodes based on the electromagnetic field theory and neglecting the electrode polarization effects. The experiments were conducted on saline solutions contained in a grounded metallic bath to validate the obtained values of the derived geometric factor. The effect of the polarization impedance at the electrodes is found to be negligible when using relatively nonpolarizable silver-silver chloride electrodes at a frequency of 3 kHz. Our experimental results also show that the resistivity values determined using both the new geometric factor and a commercial conductivity meter are in good agreement for small electrode radius, interelectrode spacing and depth of sample, therefore making it a promising technique for applications in microfluidics devices. The effects of current density and temperature on the measurement results are also presented.     

Stress concentration around a strongly oblate cavity in a cubic crystal and its associated crack problems

The stress distribution around a strongly oblate spheroidal cavity in a cubic crystal is determined by the equivalent inclusion method. The stress concentration factor is shown to be a product of two factors: one factor is purely geometric; the other factor depends on the material properties. By allowing the aspect ratio of the cavity to approach zero, the stress intensity factor of the associated penny-shaped crack is deduced. The energy release rates of cracks on {1 0 0} planes are computed for different growth directions. Theses results are found to be correlated well with Zener’s anisotropy factor.