Charged Spheroids Research Articles

Analytical analysis of anisotropic thermophoresis of a charged spheroidal colloid in aqueous media for extremely thin EDL cases.

Thermophoresis of charged spheroids has been widely applied in biology and medical science. In this work, we report an analysis of the anisotropic thermophoresis of diluted spheroidal colloids in aqueous media for extremely thin EDL cases. Under the boundary layer approximation, we formulate the thermophoretic velocity, the thermophoretic force, and the thermodiffusion coefficient of a randomly dispersed spheroid. The parametric studies show that under the aforementioned conditions, the thermophoresis is anisotropic and its thermodiffusion coefficient should be considered as a vector, DT . The thermodiffusion coefficient values and directions of DT are strongly related to the aspect ratio and the angle θ between the externally applied temperature gradient and the particle's axis of revolution: The increasing aspect ratio enlarges the thermodiffusion coefficient value DT of prolate (oblate) spheroids to a constant value when θ<60° (θ>45°), and it reduces DT of prolate (oblate) spheroids to a constant value when θ>60° (θ<45°). The thermodiffusion coefficient direction of both prolate and oblate spheroids deviates slightly from -∇T∞ for a small aspect ratio, and such deviation becomes serious for a large aspect ratio.

The interaction energy of two uniformly charged spheroids. Example of deformed nuclei

The interaction energy of two uniformly charged spheroids. Example of deformed nuclei

T-matrix formulation of electromagnetic wave scattering by charged non-spherical scatterers

In this paper, a method is proposed to solve the scattering of electromagnetic waves (EMWs) by a uniformly charged non-spherical scatterer. Based on Waterman's extended boundary method, a transition matrix between coefficient of the scattering field and incident field of a charged non-spherical scatterer is derived, called Tc-matrix. By using Tc-matrix, the scattering fields of charged non-spherical scatterers can be numerically solved. In the case of a charged rotationally symmetric non-spherical scatterer, the analytic mathematical expression of Tc-matrix is obtained, thus the analytical expression of scattering field is obtained. And the extinction characteristics of charged non-spherical scatterers, making use of charged spheroids as examples, are investigated. It is found that absorption is still the dominant mechanism of attenuation when electromagnetic waves travels through small charged spheroidal particles suspended in atmosphere, though the net surface charges can enhance the extinction/scattering/absorption of particle. The enhancement factors of extinction/absorption increases with increasing surface charge density and are much higher than the enhancement factor of scattering. It is shown that the equivalent sphere assumption, which was used in theoretical models of optical or scattering properties of charged or uncharged atmospheric or interstellar particles, could result in large errors, e.g., even over 100% for charged prolate spheroids, in theoretical prediction of absorption/extinction. Moreover, this inaccuracy due to the sphere assumption worsens for spheroids with either low or high aspect ratios h (i.e., prolate spheroid with h << 1, or oblate spheroids with h >> 1).

High-frequency dielectric response of a spheroidal particle with a thin double layer.

This paper addresses the dielectric response of a charged, oblate spheroidal particle immersed in an electrolyte. Analytic solutions are obtained for the polarization coefficient in the thin double layer limit for frequencies in the megahertz to gigahertz range. Two different distributions of the equilibrium surface charge density in the double layer are considered. The first is when the surface charge density is uniform. The second is a particular nonuniform distribution with more charges along the flat surface of the spheroid than along the edges. For this latter case, an exact solution for the polarization coefficients is obtained. In these solutions for the polarization coefficients, to a good approximation, the particles act as if they have conductivity in addition to permittivity, with different values for the conductivity depending on whether the applied electric field is parallel or perpendicular to the flat surface of the particle. When it is parallel to the flat surface, the apparent conductivity of the particle is roughly proportional to total charge on the particle and, hence, does not appear to depend on the details of the surface charge density. When the electric field is perpendicular to the flat surface, the apparent conductivity decreases as the particle becomes more platy and as the charge along the edges decreases. The effect on the dielectric dispersion of a dilute mixture of these charged spheroids in an electrolyte is also calculated. When their symmetry axes are aligned parallel to the applied electric field, the charges on the spheroids reduce the dielectric dispersion, while for spheroids aligned in the other direction, the charges increase both the conductivity and the dielectric dispersion.

Low-frequency dielectric response of charged oblate spheroidal particles immersed in an electrolyte.

We study the low-frequency polarization response of a surface-charged oblate spheroidal particle immersed in an electrolyte solution. Because the charged spheroid attracts counterions which form the electric double layer around the particle, using usual boundary conditions at the interface between the particle and electrolyte can be quite complicated and challenging. Hence, we generalize Fixman's boundary conditions, originally derived for spherical particles, to the case of the charged oblate spheroid. Given two different counterion distributions in the thin electric double-layer limit, we obtain analytic expressions for the polarization coefficients to the first nontrivial order in frequency. We find that the polarization response normal to the symmetry axis depends on the total amount of charge carried by the oblate spheroid while that parallel to the symmetry axis is suppressed when there is less charge on the edge of the spheroid. We further study the overall dielectric response for a dilute suspension of charged spheroids. We find that the dielectric enhancement at low frequency, which is driven by the presence of a large ζ potential (surface charge), is suppressed by high ion concentrations in the electrolyte and depends on the size of the suspended particles. In addition, spheroids with higher aspect ratios will also lead to a stronger dielectric enhancement due to the combination of the electric double layer and textural effects. The characteristic frequency associated with the dielectric enhancement scales inversely with the square of the particle size, the major radius of the spheroid, and it has a weak dependence on the shape of spheroids.

Analysis on electrostatic behavior of a conducting prolate spheroid under an electric field

This paper presents the electrostatic analysis of a conducting prolate spheroid under an external electric field in axisymmetric configurations. The configurations consist of a conducting spheroid which is in contact with or separated from a grounded plane. We apply the method of images using multipoles to the electric field calculation. The calculation uses the multipole re-expansion formulae and the equivalent image charges of a prolate conducting spheroid. The induced charge and force are determined from the multipole images. For a spheroid on the grounded plane, the calculation results show that the maximal field increases nonlinearly with the major-to-minor axis ratio of the spheroid. The charge and the force are compared between the spheroids of the same major axis or the same surface area. We propose empirical formulae for approximating the maximal field, charge and force with the errors smaller than 2% for the axis ratio between 1 and 32. We examine the propriety of the approximation of the electric field, charge and force based on a hemispheroidal model. For an uncharged spheroid separated from the grounded plane, we clarify the relationship between the separation and the field intensification at the bottom pole of the spheroid. Based on the critical field strength of a background medium, the position of the spheroid where the partial discharge takes place is estimated. For a charged spheroid, we found that the maximal electric field hardly depends on the separation from the plane.

Coulomb explosion of uniformly charged spheroids

A simple, semianalytical model is proposed for nonrelativistic Coulomb explosion of a uniformly charged spheroid. This model allows us to derive the time-dependent particle energy distributions. Simple expressions are also given for the characteristic explosion time and maximum particle energies in the limits of extreme prolate and oblate spheroids as well as for the sphere. Results of particle simulations are found to be in remarkably good agreement with the model.

Depletion Interactions Produced by Nonadsorbing Charged and Uncharged Spheroids

The effect of macromolecule shape on the depletion attraction between two hard spherical particles in a solution with nonadsorbing hard spheroidal macromolecules of arbitrary size and aspect ratio was investigated using a modified form of the force-balance model of J. Y. Walz and A. Sharma (1994, J. Colloid Interface Sci.168, 495). The macromolecules were represented as general spheroids, which could be either charged or uncharged. For the uncharged case, a set of analytical expressions describing the depletion attraction, valid for particles much larger than the characteristic macromolecule size, was developed. Comparisons with the case of spherical macromolecules were made under the condition of either constant macromolecule number density, ρb, or constant volume fraction, φ. It was found that increasing the spheroidal macromolecule aspect ratio (major axis length/minor axis length) decreases the depletion attraction at constant ρb, but increases the interaction at constant φ. In the latter case, the interaction produced by prolate macromolecules is greater than that produced by oblate macromolecules of equal axis lengths, while the opposite is true at constant ρb. A simple scaling analysis is used to explain these trends. Surface charge is found to increase both the range and the magnitude of the depletion attraction; however, the general trends are the same as those found in the uncharged systems. Finally, the effect of the depletion attraction produced by spherical and spheroidal macromolecules on the stability of a dispersion of charged particles was examined. It was found that charged spheroids at concentrations of order 1% volume can produce secondary energy wells of sufficient magnitude to induce flocculation in a dispersion of charged spherical particles.

Electrical Interactions between Two Charged Spheroids in a Symmetric Electrolyte Solution

The interactions between two charged spheroids immersed in a symmetric electrolyte solution are evaluated. We show that the more closely two particles resemble spheres, the higher the electrical interaction energy between them. The variation in the total interaction energy between two particles, which includes the electrical and the van der Waals energies, reveals that the greater the radius of curvature of a particle, the higher the primary maximum or the harder it is for two particles to undergo an irreversible coagulation. If the double layer around a particle is thin compared with its radius of curvature, the critical coagulation concentration (CCC) of counterions is independent of the latter. This implies that the relative orientation between two particles is insignificant as far as the CCC is concerned. In this case, the result obtained in the present study reduces to that predicted by the Derjaguin−Landau−Verwey−Overbeek theory.

Electrostatic Interaction between Two Ion-Penetrable Charged Spheroids

The electrostatic interaction between two ion-penetrable, charged spheroidal particles is examined theoretically. These particles can assume different sizes and an arbitrary spatial orientation. The electrical potential distribution is derived analytically under the Debye–Huckle condition. The results for two interaction spheres, one spheroidal particle and a planar surface, and rigid particles covered by an ion-penetrable membrane can be recovered as the special cases of the present general problem. We show that, for a fixed center-to-center distance between two particles, regardless of their relative sizes, the interaction free energy is the greatest if their major axes lie on the same line (head-to-head), and the smallest if their major axes are perpendicular to each other but not on the same plane (perpendicular).

Electrophoretic transport of spheroidal colloids in nonhomogeneous electric fields

The velocity of a uniformly charged spheroid in a spatially varying electric field, E∞(x), is determined. The electrical double layer at the particle's surface is assumed to be very thin relative to the size of the particle. The electric field produces a slip velocity at the surface of the particle that drives the electrophoretic motion. If the particle is not a sphere, then its velocity is not given simply by Smoluchowski's equation. The translational velocity of the particle is proportional to E∞ and ⊇⊇E∞, while the angular velocity is proportional to ⊇E∞. Examples are presented to demonstrate the effectiveness of nonhomogeneous fields in orienting particles with and without the opposing effects of Brownian rotation. It should be possible to design electrode configurations that would produce the nonhomogeneous electric fields necessary to align and transport colloidal particles in a desired manner.

On the boundary-value problem of a spheroid

The surface charge density of a charged spheroid is obtained in exact, closed form using a Green’s function expansion in spherical coordinates. The possibility is thus established of solving boundary-value problems analytically using coordinates that do not correspond to boundary shapes. The present approach, used previously in numerical solutions of related problems, requires the potential function to be constant in the interior of the conductor. Its advantages from the theoretical standpoint and its further possibilities are discussed.

Charged Spheroid in Cylinder

The problem of a charged conducting spheroid within a coaxial conducting cylinder is solved by a slight variation of the method in J. Appl. Phys. 31, 553 (1960). Errors in the terminal digits of Table I in that paper have been corrected and the table extended. The charge density on the spheroid, the potential between it and the cylinder, and its capacitance are given for ratios of the spheroid equatorial radius to cylinder radius of 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, and 0.95. Tables give numerical results for five cases: the disk, the oblate spheroid with 2 to 1 axial ratio, the sphere, and the prolate spheroid with axial ratio 1 to 2. The thin prolate spheroid requires special treatment.