Shape factor hydrodynamic

Agglomeration in a slurry causes a change in the packing factor at which flow is blocked, from 0.6 to 0.45 and causes a change in the hydrodynamic shape factor from 3.5 to 2.5. In both cases, the volume fraction of dispersed particles is 0.40. 

Here, fp is once again the volnme fraction of dispersed spheres, and /Ch is the apparent hydrodynamic shape factor, bnt a new parameter, /., is introduced which accounts... 

Figure 4.7 Effect of hydrodynamic shape factor on effective particle size in a flow field. From J. S. Reed, Principles of Ceramics Processing, 2nd ed. Copyright 1995 by John Wiley Sons, Inc. This material is used by permission of John Wiley Sons, Inc.

Other biological fluids can be modeled as suspensions of particles in a solvent, such as was used for the description of suspensions and slurries in Section 4.1.2.2, namely, that the relative viscosity of the suspension is related to the hydrodynamic shape factor. 

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. 

Scheraga-Mandelkern equations (1953), for effective hydrodynamic ellipsoid factor p (Sun 2004), suggested that [rj] is the function of two independent variables p, the axial ratio, which is a measure of shape, and Ve, the effective volume. To relate [r ] to p and Ve, introduced f, the frictional coefficient, which is known to be a direct function of p and Ve. Thus, for a sphere we have... 

Poly(/ -phenylethyl isocyanide) was similarly prepared and fractionated (14). A comparison between the hydrodynamic properties of poly(/T and poly(a-phenylethyl isocyanide) showed, that while the latter was characterized by its intrinsic lack of molecular flexibility, the former was relatively a flexible chain. This was manifested in the values estimated for the shape factor and the radius of gyration. Accordingly, two general conformations in dilute solution are ascribed to poly(phenylethyl isocyanides) a nearly rigid, rodlike helix to poly(a-phenyl-ethyl isocyanide), and an undulating, more randomly orienting chain to poly(/l-phenylethyl isocyanide). 

It is apparent, from Eq. (1), that the primary sample property measured by flow FFF is the diffusion coefficient. Secondary information includes the hydrodynamic diameter which can be obtained via the Stokes-Einstein equation and the molecular weight if the molecule shape factor is constant. Unlike other FFF techniques, the retention time in flow FFF is determined solely by the diffusion coefficient rather than a combination of sample properties. As a consequence, flow FFF is well suited for analyses of complex sample mixtures and the transformation of the fractogram to a diffusion or size distribution is straightforward. In addition, flow FFF is applicable to a wide range of samples regardless of their charge, size, density, and so forth. 

Sedimentation coefficients of protein-surfactant complexes, subunit dissociation and molecular weights [26,27,43] Hydrodynamic volume and shape factors, protein unfolding [38,39]... 

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. 

The shape of a suspended drop depends on the surface tension as well as on the gravitational force. The drop is photographed and the diameter at various positions is measured. A consistent shape factor can be evaluated when hydrodynamic equilibrium is reached. 

Light scattering (LS) provides information related to the dimensions of the polyplexes (hydrodynamic radius f h), their shape (radius of gyration f g and shape factor p = / g/ h)> well as weight-average molecular weight (Mw) of the aggregates and polydispersity of the sample. 

Comparing Eqs. (83), (84) and Eqs. (21), (22) it follows immediately that Rouse and Zimm relaxation result in completely different incoherent quasielastic scattering. These differences are revealed in the line shape of the dynamic structure factor or in the (3-parameter if Eq. (23) is applied, as well as in the structure and Q-dependence of the characteristic frequency. In the case of dominant hydrodynamic interaction, Q(Q) depends on the viscosity of the pure solvent, but on no molecular parameters and varies with the third power of Q, whereas with failing hydrodynamic interaction it is determined by the inverse of the friction per mean square segment length and varies with the fourth power of Q. 

---