Particles ellipsoidal

The viscosity of a suspension of ellipsoids depends on the orientation of the particle with respect to the flow streamlines. The ellipsoidal particle causes more disruption of the flow when it is perpendicular to the streamlines than when it is aligned with them the viscosity in the former case is greater than in the latter. For small particles the randomizing effect of Brownian motion is assumed to override any tendency to assume a preferred orientation in the flow. 

Rigid particles other than unsolvated spheres. It is easy to conclude qualitatively that either solvation or ellipticity (or both) produces a friction factor which is larger than that obtained for a nonsolvated sphere of the same mass. This conclusion is illustrated in Fig. 9.10, which shows the swelling of a sphere due to solvation and also the spherical excluded volume that an ellipsoidal particle requires to rotate through all possible orientations. 

Figure 9.10 Schematic relationship between the radius Rq of an unsolvated sphere and the effective radius R of a solvated sphere or of a spherical volume excluded by an ellipsoidal particle rotating through all directions.

The representation of the dispersion viscosity given above related to suspensions of particles of spherical form. Upon a transition to anisodiametrical particles a number of new effects arises. The effect of nonsphericity is often discussed on the example of model dispersions of particles of ellipsoidal form. An exact form of particles to a first approximation is not very significant, the degree of anisodiametricity is only important, or for ellipsoidal-particles, their eccentricity. 

Returning to Staudinger s derivation, it must be revised on two grounds. First the kinematics of motion is three rather than two-dimensional and the hydrodynamic volume spherical rather than cylindrical, i.e. 3. The detailed calculation for thin ellipsoidal particles (13) shows an approximate proportionality of the intrinsic viscosity with M1 7, a considerable difference from eq. (1) for large M. 

Theory. In the general case where rigid revolution ellipsoidal particles in solution possess both a permanent and an induced dipolar moment colinear with the particle optical axis, the theory derived by Tinoco predicts the following behaviour of the solution birefringence An(t) in the limit of weak electric field (6). 

There are some other conditions that may affect the appearance of hematite. For example, ellipsoidal particles of various anisometries were obtained by the addition of small amounts of phosphate ions into the aging FeCl3 solutions (71). A possible explanation for the effect of this anion was suggested in Refs. 100 and 143. 

Fig. 5.2.6 SEM images of the ellipsoidal particles of basic aluminum sulfate prepared in the presence of 1.67 mol dm-3 chloride ions under the otherwise standard conditions. (From Ref. 1.)...

The hydrodynamic shape factor and axial ratio are related (see Eigure 4.18), but are not generally used interchangeably in the literature. The axial ratio is used almost exclusively to characterize the shape of biological particles, so this is what we will utilize here. As the ellipsoidal particle becomes less and less spherical, the viscosity deviates further and further from the Einstein equation (see Eigure 4.19). Note that in the limit of a = b, both the prolate and oblate ellipsoid give an intrinsic viscosity of 2.5, as predicted for spheres by the Einstein equation. 

The natural coordinates, albeit unfamiliar and not without their disagreeable features, for formulating the problem of determining the dipole moment of an ellipsoidal particle induced by a uniform electrostatic field are the ellipsoidal coordinates ( , rj, f) defined by... 

Only when the particle is in free space (em = e0) are its depolarization factors independent of composition in this instance = Lj/e0. The induced field in an ellipsoidal particle is uniform but not necessarily parallel to an arbitrary applied field except in the special case of a sphere. 

It might seem at first glance that arriving at the dipole moment p of an ellipsoidal particle via the asymptotic form of the potential < p is a needlessly complicated procedure and that p is simply t>P, where v is the particle volume. However, this correspondence breaks down for a void, in which P, = 0, but which nonetheless has a nonzero dipole moment. Because the medium is, in general, polarizable, uP, is not equal to p even for a material particle except when it is in free space. In many applications of light scattering and absorption by small particles—in planetary atmospheres and interstellar space, for example—this condition is indeed satisfied. Laboratory experiments, however, are frequently carried out with particles suspended in some kind of medium such as water. It is for this reason that we have taken some care to ensure that the expressions for the polarizability of an ellipsoidal particle are completely general. 

Although the necessary labor is increased, no new concepts are required to extend the results above for a homogeneous ellipsoid to a coated ellipsoid. We denote by el the permittivity of the inner or core ellipsoid with semiaxes flj, c, e2 is the permittivity of the outer ellipsoid with semiaxes a2,b2,c2. This coated ellipsoidal particle is in a medium with permittivity em. As in the preceding section, we introduce ellipsoidal coordinates , tj, f ... 

More general ellipsoidal particles in an anisotropic medium, where there is no restriction on the principal axes of either the real or imaginary parts of the permittivity tensors, have been treated by Jones (1945). 

For any real material, the frequency at which (12.27) is satisfied is complex—the surface modes are virtual. However, its real part is approximately the frequency where the cross sections have maxima, provided that the imaginary part is small compared with the real part. We shall denote this frequency by us. For a sphere, o>s is the Frohlich frequency wF. If used intelligently, always keeping in mind its limitations, (12.27) is a guide to the whereabouts of peaks in extinction spectra of small ellipsoidal particles but it will not necessarily lead to the exact frequency. 

In the preceding paragraphs we discussed only the conditions for surface mode resonances in the cross sections of small ellipsoidal particles. We now turn to specific examples to further our understanding of these resonances. 

Up to this point we have considered only a single ellipsoidal particle oriented so that the electric field of the incident wave is parallel to one of its... 

At this point the reader who has studied the preceding sections may well wonder why we are so interested in, if not obsessed with, ellipsoidal particles most real particles are no more ellipsoidal than they are spherical. One reason for devoting so much space to ellipsoids is that they are a means for dispelling widespread misconceptions about the nonexistence of shape effects in small-particle absorption spectra. For if there are strong shape effects in spectra of ellipsoidal particles, then there are certainly such effects in the spectra of other, less well-defined nonspherical particles. But there is at least one other reason, which may prove to be of greater practical utility the hope that spectra of ii regular particles can be approximated somehow by suitably averaging over all... 

Gilra (1972ab) has very thoroughly discussed absorption by small ellipsoidal particles, including those with coatings, in the surface mode region. 

Bohren, C. F., and D. R. Huffman, 1981. Absorption cross-section maxima and minima in IR absorption bands of small ionic ellipsoidal particles, Appl. Opt., 20, 959-962. 

FIG. 1.14 Model particles of different shapes with the same or different chemical compositions (a) rodlike particles of akageneite (/3-FeOOH) (b) ellipsoidal particles of hematite (a-Fe203) (c) cubic particles of hematite and (d) rodlike particles of mixed chemical composition (a-Fe203 and /3-FeOOH). All are TEM pictures. (Reprinted with permission of Matijevic 1993.)... 

The final surfactant structures we consider as models for biological membranes are vesicles. These are spherical or ellipsoidal particles formed by enclosing a volume of aqueous solution in a surfactant bilayer. When phospholipids are the surfactant, these are also known as liposomes, as we have already seen in Vignette 1.3 in Chapter 1. Vesicles may be formed from synthetic surfactants as well. Depending on the conditions of preparation, vesicle diameters may range from 20 nm to 10 pirn, and they may contain one or more enclosed compartments. A multicompartment vesicle has an onionlike structure with concentric bilayer surfaces enclosing smaller vesicles in larger aqueous compartments. 

Exercise. The rotation of an ellipsoidal particle suspended in a fluid obeys the macroscopic equation of motion... 

Applications of optical methods to study dilute colloidal dispersions subject to flow were pioneered by Mason and coworkers. These authors used simple turbidity measurements to follow the orientation dynamics of ellipsoidal particles during transient shear flow experiments [175,176], In addition, the superposition of shear and electric fields were studied. The goal of this work was to verify the predictions of theories predicting the orientation distributions of prolate and oblate particles, such as that discussed in section 7.2.I.2. This simple technique clearly demonstrated the phenomena of particle rotations within Jeffery orbits, as well as the effects of Brownian motion and particle size distributions. The method employed a parallel plate flow cell with the light sent down the velocity gradient axis. 

Jefferey, The motion of ellipsoidal particles immersed in a viscous fluid, Proc. R. Soc. London, A, 102, 161 (1922). 

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