Nearly Spherical Particle

So far we have treated uniformly charged planar, spherical, or cylindrical particles. For general cases other than the above examples, it is not easy to solve analytically the Poisson-Boltzmann equation (1.5). In the following, we give an example in which one can derive approximate solutions. 

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. 

Consider first a prolate spheroid with a constant uniform surface potential ij/f, in an electrolyte solution (Fig. 1.17a). The potential j/ is assumed to be low enough to obey the linearized Poisson-Boltzmann equation (1.9). We choose the z-axis as the axis of symmetry and the center of the prolate as the origin. Let a and b be the major and minor axes of the prolate, respectively. The equation for the surface of the prolate is then given by 

when the spheroid is nearly spherical (i.e., for low Eq. (1.207) becomes 

FIGURE 1.17 Prolate spheroid (a) and oblate spheroid (b). a and b are the major and minor semiaxes, respectively. The z-axis is the axis of S3fmmetry. 

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Molerus and Schweinzer is given by the following equation for spherical (or nearly spherical) particles. 

Rowe, P.N., Nienow, A.W. and Agbim, A.J., The mechanisms by which particles segregate in gas fluidised beds binary systems of near-spherical particles, Trans. Inst. Chem. Engrs., 50 (1972a) 310-323. 

In the point matching method (Oguchi, 1973 Bates, 1975) the fields inside and outside a particle are expanded in vector spherical harmonics and the resulting series truncated the tangential field components are required to be continuous at a finite number of points on the particle boundary. Although easy to describe and to understand, the practical usefulness of this method is limited to nearly spherical particles large demands on computer time and uncertain convergence are also drawbacks (Yeh and Mei, 1980). 

The shapes of the absorption band cease to be independent of size for particles smaller than about 26 A, which suggests that the bulk dielectric function is inapplicable. Indeed, the broadening and lowering of the absorption peak can be explained by invoking a reduced mean free path for conduction electrons (Section 12.1). Thus, the major features of surface modes in small metallic particles are exhibited by this experimental system of nearly spherical particles well isolated from one another. But when calculations and measurements with no arbitrary normalization are compared, some disagreement remains. Measurements of Doremus on the 100-A aqueous gold sol, which agree with those of Turkevich et al., are compared with his calculations in Fig. 12.18 the two sets of calculations are for optical constants obtained... 

An ultrathin section of a resin inclusion in vitrain from a subbituminous coal is shown in Figure 4. This variety of resin appeared to have a very fine granular structure in the light microscope. The micrograph shows that this consists of nearly spherical particles ranging from 300 to 700 A. in diameter. 

The easiest model to treat theoretically is the sphere, and many colloidal systems do, in fact, contain spherical or nearly spherical particles. Emulsions, latexes, liquid aerosols, etc., contain spherical particles. Certain protein molecules are approximately spherical. The crystallite particles in dispersions such as gold and silver iodide sols are sufficiently symmetrical to behave like spheres. 

The governing heat transfer modes in gas-solid flow systems include gas-particle heat transfer, particle-particle heat transfer, and suspension-surface heat transfer by conduction, convection, and/or radiation. The basic heat and mass transfer modes of a single particle in a gas medium are introduced in Chapter 4. This chapter deals with the modeling approaches in describing the heat and mass transfer processes in gas-solid flows. In multiparticle systems, as in the fluidization systems with spherical or nearly spherical particles, the conductive heat transfer due to particle collisions is usually negligible. Hence, this chapter is mainly concerned with the heat and mass transfer from suspension to the wall, from suspension to an immersed surface, and from gas to solids for multiparticle systems. The heat and mass transfer mechanisms due to particle convection and gas convection are illustrated. In addition, heat transfer due to radiation is discussed. 

Besides particle size and dmg concentration, particle shape and density of powders are also important factors for achieving a homogenous mixture. Powders with nearly spherical particles are easier to mix than those with irregularly shaped particles. Micronized, needle, or flat particles require longer blending time due to aggregate... 

Rowe, P. N., Nienow, A. W., and Agbim, A. J., The Mechanism by Which Particles Segregate in Gas Fluidized Beds—Binary Systems of Near-Spherical Particles, Trans, lnstn. Chem. Engrs. 50,310(1972a). 

This is expected assuming the ideal homogeneity of both the incident beam and the sample packing density. The former is true for small divergence slit openings, and the latter is true for the used sample, which was prepared from the nearly spherical particles. 

These few examples do not exhaust a long list of difficulties in calculation of exact values of potential from electrophoretic or electroacoustic data, and most results reported in literature appear to be rough estimates rather than exact values, except for some results obtained with nearly spherical particles having very narrow size distributions, e.g. Stober silica. Many publications tend to overestimate the accuracy of ( potentials presented therein. For example in a recent publication a graph was presented showing the variations of the measured potential as a function of time, and all data points showed in that graph ranged from 6 to 7 mV. 

It is not surprising therefore that the optical properties of small metal particles have received a considerable interest worldwide. Their large range of applications goes from surface sensitive spectroscopic analysis to catalysis and even photonics with microwave polarizers [9-15]. These developments have sparked a renewed interest in the optical characterization of metallic particle suspensions, often routinely carried out by transmission electron microscopy (TEM) and UV-visible photo-absorption spectroscopy. The recent observation of large SP enhancements of the non linear optical response from these particles, initially for third order processes and more recently for second order processes has also initiated a particular attention for non linear optical phenomena [16-18]. Furthermore, the paradox that second order processes should vanish at first order for perfectly spherical particles whereas experimentally large intensities were collected for supposedly near-spherical particle suspensions had to be resolved. It is the purpose of tire present review to describe the current picture on the problem. 

One specialised method of particle agglomeration is extrusion and spheronisation, to produce spherical or near-spherical particles. Such particles are suitable for coating with release modifying coats to produce controlled release formulations. The particles are usually filled into hard gelatin capsules for administration to patients. The process of extrusion/spheronisation is depicted in Figure 11.22. 

Practically, this method works well for spherical or near-spherical particles however, it produces erroneous results for nonspherical particles. Moreover, this method is not suitable for porous material because the effective particle density is not known and the volume measured is the envelope volume. Special care is required to avoid crowding of the orifice otherwise, special treatment is needed to analyze the instrument counts. 

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