Effective hydrodynamic ellipsoid volume

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... 

In the hydrodynamic theories for any of the quantities in the last column of Table I, each quantity depends on two parameters, the size and the shape. For example, for an ellipsoid of revolution these two parameters could be the volume and axial ratio, F and p, respectively, of the effective hydrodynamic ellipsoid. Therefore, we may draw a first very important conclusion, viz., that a determination of only one quantity, e.g., /, cannot provide us with a value of Ve or p. It is clear that two quantities are required, e.g., f and [r ] if two such quantities are available, then both F, and p may be computed. Of course, if one has information in advance as to the value of F or p (e.g., if the particle behaves as a sphere, with p = 1), then a single hydrodynamic quantity suffices to give the value of the other parameter. Unfortunately, one never has this advance information and, therefore, must carry out two different kinds of hydrodynamic measurements in order to obtain a pair of hydrodynamic quantities. 

Again, fo refers to a sphere of the same volume as the effective hydrodynamic ellipsoid and 1/J is a function of p only (Perrin, 1934). 

The compaction behavior is also indicated by the effective volume fraction of an aggregate (pejf (Fig- 5), calculated from the aspect ratio of an equivalent ellipsoid. The shear-rale dependence on effective volume fraction was small so that a maximum compaction was not observed even at high shear rates. Again, higher shear rates required for higher compaction may violate model assumptions, such as Stokes regime for the hydrodynamics and the conditions for the overdamped motion. 

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