Shape factor ellipsoid

The biggest difference between biological particles and ceramic particles in the application of Eq. (4.20) is that while most ceramic particles are spherical ( Ch = 2.5), most biological particles can be modeled as either prolate ellipsoids or oblate spheroids (or ellipsoids). Ellipsoids are characterized according to their shape factor, ajb, for which a and b are the dimensions of the semimajor and semiminor axes, respectively (see Eigure 4.17). In a prolate ellipsoid, a > b, whereas in an oblate ellipsoid, b > a.ln the extremes, b approximates a cylinder, and b a approximates a disk, or platelet. 

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. 

Figure 4.18 Theoretical values of the shape factor for ellipsoids. Adapted from F. H. Silver and D. L. Christiansen, Biomaterials Science and Biocompatibility, p. 150. Copyright 1999 by Springer-Verlag.

However, macromolecular solutes are seldom perfectly spherical their shapes are more often best approximated as ellipsoids of revolution, for example, prolate (elongated spindle-shaped) and oblate (flattened disc-shaped) ellipsoids. The shape factor... 

Figure 4.1. Theoretical values of the shape factor for ellipsoids. This is a plot of the shape factor, log v versus log of the axial ratio alb) for prolate and oblate ellipsoids. Note that intrinsic viscosity can be approximated by the shape factor for ellipsoids.

Because collagen fibrils are thin long elements, their shape factors can be estimated from the equations used for prolate ellipsoids (see Figure 4.1). For prolate ellipsoids the shape factor is equal to a constant times the ratio of the major semiaxis length a, divided by the minor semiaxis length b, raised to power of 1.81 (Equation (8.5)). 

Shape Factors for the Calculations of Lengths of Prolate Ellipsoids... 

Figure 4.1. Shape factor ratio against perimeter-equivalent factor for particles of various shape in a stagnant medium 1, circular cylinder 2, oblate ellipsoid of revolution 3, prolate ellipsoid of revolution 4, cube...

Shape Factors for Ellipsoids Integral Form for Numerical Calculations... 

The capacity and/or the capacitance of an isopotential ellipsoid are presented in several texts and handbooks such as those by Flugge [22], Jeans [40], Kellogg [43], Mason and Weaver [62], Morse and Feshbach [68, 69], Smythe [98], and Stratton [111]. The results presented in these texts are used to develop expressions for the shape factors of several bodies spheres, oblate and prolate spheroids (see Fig. 3.3), circular and elliptical disks, and ellipsoids. The shape factor for the ellipsoid is general it reduces to the shape factor for the other bodies. 

Dimensionless Shape Factor and Diffusion Length of Ellipsoids. The dimensionless shape factor. S vT and the proposed dimensionless diffusion length VA/A for isothermal ellipsoids can be obtained from the shape factor integral 7((3, y) and the relationship VA/a given previously. The equation is... 

The depolarization factor L in equation (13) as well as the depolarization factors Li in equation (14) can be calculated from the mean aspect ratio Q of the particles as well as from the mean shape factor S. For ellipsoids with three doubled half-axes Da, L, and Dc, Li is given by equation (15) [23]. 

In a similar approach, double-stranded helicates of various lengths that were derived from copper and silver-based metallosupramolecular architectures have also been classified by their diffusion properties and estimates of the molecular sizes made [48]. Owing to the ellipsoidal structures, it was necessary to introduce appropriate shape factors to translate the hydrodynamic radii determined directly from the unmodified Stoke-Einstein equation into dimensions that were meaningful for these assemblies. Thus, knowledge of the width of the helicates (determined from the X-ray structure of a single complex in this case) allowed the determination of their lengths from the hydrodynamic radii. The results for a series of these helicates is summarised in Table 9.9. It was further shown that 2D DOSY spectra could be employed to differentiate the helicates of different lengths when present simultaneously in a mixture. 

Surface shape factors are much more difficult to measure than volume shape factors and they are subject to greater uncertainty. One method is as follows. A few individual crystals are observed through a low-power microscope fitted with a calibrated eyepiece, and sufficient measurements taken to allow a sketch to be made of a representative geometric shape, e.g. a parallelepipedon, ellipsoid, oblate spheroid, etc. The surface area of the representative solid body may then be calculated. It should be appreciated, of course, that the result of such a calculation will be prone to significant error. 

Here, s, Sgm, and s are the dielectric functions of the particles, the surrounding, and the effective medium, respectively, and gk (k = 1, 2, 3) is the geometric (shape) factor, which determines the self-polarizing effect of the ellipsoid and has a value between zero (needle) and unity (slab) so that gi -h g2 + gs = 1 ... 

In the more general case, which is more complicated than the bilayer dielectric, the components are modelled by ellipsoidal elements, and the shape factors ascribed to the phases are also to be taken into account. One can use one- and two-phase elemntary units in the calculations [3]. 

Mie theory gives the rigorous solutions of wave equations only for spherical nanoparticles. For particles with other shapes like ellipsoids, Mie theory cannot be applied, and treatment with the dipolar approximation is useful to discuss the optical properties of the particle qualitatively. By taking the shape-dependent depolarization factor into account, the polarizability of the ellipsoid can be obtained as a form similar to that of the sphere [42]. 

The shape factor /s is always greater than 1 and can be theoretically computed for prolate and oblate ellipsoids... 

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