Spherical/spheroid particles

We give below a simple method to derive an approximate solution to the hnear-ized Poisson-Boltzmann equation (1.9) for the potential distribution i/ (r) around a nearly spherical spheroidal particle immersed in an electrolyte solution [12]. This method is based on Maxwell s method [13] to derive an approximate solution to the Laplace equation for the potential distribution around a nearly spherical particle. 

The Stokes-Einstein equation has already been presented. It was noted that its vahdity was restricted to large solutes, such as spherical macromolecules and particles in a continuum solvent. The equation has also been found to predict accurately the diffusion coefficient of spherical latex particles and globular proteins. Corrections to Stokes-Einstein for molecules approximating spheroids is given by Tanford. Since solute-solute interactions are ignored in this theory, it applies in the dilute range only. 

Depending on operation conditions and metal properties, the shapes of the atomized particles may be spheroidal, flaky, acicular, or irregular, but spherical shape is predominant. The spheroidal particles are coarse. For example, roller-atomized Sn particles exhibited a mass median diameter of 220 to 680 pm. The large particle sizes and highly irregular particle shapes suggested that the disintegration process may be arrested either by the premature solidification or by the formation of a thick, viscous oxide layer on the liquid surface. The particle size distributions were found to closely follow a log-normal pattern even for non-uniform particle shapes. 

Keh and Chen [33] employed O Brien s method [7] to examine the polarization effect on the electrophoresis of an infinitely long circular cylinder. They found that neglecting the end effect, the transverse electrophoretic velocity is identical to that for a spherical particle with the same radius. The polarization effects were also investigated for a spheroidal particle [34,35] and an infinitely long elliptical cylinder [36]. An interesting feature discovered from these studies is that the electrophoretic velocity decreases with the reduction of the maximum length of the particle in the direction of the migration. 

Perfect spheres are rare, but spheroidal particles are present in some powders produced at high temperature (e.g. pyrogenic silicas) or by the sol-gel process. The term sphericity is useful for some purposes. Sphericity has been defined in various ways, the simplest definition being the ratio of the surface area of a sphere of the same volume as a given particle to the actual surface area of that particle (Allen, 1990). 

Non-porous Zr02 powders can be produced by high-temperature vapour phase condensation methods in this manner discrete spherical particles of c. 4 nm diameter have been obtained (Avery and Ramsay, 1973). It is also possible to prepare colloidal dispersions of sub-micron sized, spheroidal particles of basic salts such as Zr2(OH)6C03 and Zr2(0H)6SO4 with the aid of the carefully controlled sol-gel techniques developed by Matijevic (1988). 

An extension of the use of RF plasma for particle heating is the spheroidization of solids. By careful control of plasma enthalpy, particle size, feed rate, and feed position, it is possible to melt each particle as it passes through the plasma. The liquid droplet forms a sphere due to surface tension and, on cooling, retains its spherical shape. Spheroidized particles are commercially useful because they will flow easily. 

Particles derived from the evaporation of solution droplets are spheroidal. Shape (primarily in surface features), density, and size control of particles can be achieved by the appropriate selection of the compound, the concentration of the solution, the size of the droplet generated, and the conditions for the evaporation of the droplets. Fast evaporation rates tend to produce less solid and rough-surface particles, but this is tempered by the chemical properties of the compound. Smooth, spherical particles call for compounds with high solubility and slow evaporation rates. These requirements were used by Vanderpool and Rubow [48] to produce solid, smooth spheres of up to 70 pm in diameter. The different types of particles that can be produced from the evaporation of solution droplets include solid spheres with surfaces that are smooth, cracked, or wrinkled hollow spheres, shells, and spheroidal particles that have a wrinkled surface like raisins porous-type particles that are perforated with holes, and single crystals and particles composed of several crystals, which may be angular or spheroidal in shape. 

For example, a novel, versatile technique for the synthesizing of uniform hollow capsules from a broad range of materials is based on a combination of colloidal templating and self-assembly processes [11.8]. Fig. 11.11 describes schematically the concept. Colloidal templates of different composition, size, and geometry (although spheroidal shape is preferred) can be employed. Materials range from spherical polymer particles to non-spherical biocolloids, all vdth diameters in the nano to micrometer range. The... 

Particle shape (SEM) Spherical Spherical Spheroidal Spheroidal... 

FIG. 28 Various monolayers near the jamming state generated in the RSA simulations (a) spherical particles at homogeneous surfaces (b) spherical particles at precovered smfaces (c) spheroidal particles (ellipses) adsorbing side-on (d) spherocylinders adsorbing side-on (e) prolate spheroids (unoriented adsorption) (f) oblate spheroids (unoriented adsorption). 

The final droplet/particle shape is determined by the time required for a deformed droplet to convert to spherical shape under surface tension force. If a droplet solidifies before the surface tension force contracts it into a sphere, the final droplet shape will be irregular. Nichiporenko and Naida[488l proposed the following dimensionally correct expression for the estimation of the spheroidization time, tsph ... 

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