Translational diffusion coefficient, ellipsoidal particles

Tanford [13] and Perrin [14] determined the translational diffusion coefficient for ellipsoidal particles ... 

As noted above, for ellipsoidal particles, determination of the dimensions L and d from [r]] data via Eqs. (1.32) to (1.35) requires prior knowledge of the molecular hydrodynamic volume [Eq- (1-31)], which requires information on the partial specific volume and the bound solvent. This requirement can be circumvented by combining information on two hydrodynamic properties for example, the intrinsic viscosity and the translational diffusion coefficient in the limit c -> 0, D . Thus, the Elory-Mandelkern coefficient, ySpM. given by... 

Detailed molecnlar dynamics simnlations have been carried out also for GB particles in the sea of spheres. These studies have indicated anisotropic diffusion for the ellipsoids at higher density. In addition, the ratio between parallel and perpendicular diffusion coefficients rises from unity to the value of aspect rado as density of the system increases [19-21]. The product of the translational diffusion coefficient and reorientational correlation time behaves in a manner similar to that found for pure GB fluid. 

Stokes-Einstein Relationship. As was pointed out in the last section, diffusion coefficients may be related to the effective radius of a spherical particle through the translational frictional coefficient in the Stokes-Einstein equation. If the molecular density is also known, then a simple calculation will yield the molecular weight. Thus this method is in effect limited to hard body systems. This method has been extended for example by the work of Perrin (63) and Herzog, Illig, and Kudar (64) to include ellipsoids of revolution of semiaxes a, b, b, for prolate shapes and a, a, b for oblate shapes, where the frictional coefficient is expressed as a ratio with the frictional coefficient observed for a sphere of the same volume. 

In addition to translational Brownian motion, suspended molecules or particles undergo random rotational motion about their axes, so that, in the absence of aligning forces, they are in a state of random orientation. Rotary diffusion coefficients can be defined (ellipsoids of revolution have two such coefficients representing rotation about each principal axis) which depend on the size and shape of the molecules or particles in question28. 

Rotational Brownian motion results in the disordering of anisometric particles previously oriented in some particular way owing to the flow of the dispersion medium (see Chapter IX) or the application of an electric field. From the time of the disordering one can determine the rotational diffusion coefficient, and, for known particle size and shape, also Avogadro s number. If the particles are able to undergo co-orientation, they usually are of substantially anisometric shape, and their translational and rotational diffusion coefficients differ from those obtained for spherical particles. For example, for prolate ellipsoids of revolution with a ratio of their main diameters d] d2-= 1 10, the diffusion coefficient, D, is about 2/3 of the value obtained for spherical particles of the same volume. 

Figure 3.4 Translational dilfusion of elliptical particles. Variation of the fiictional ratio (///o) for translational diffusion of a prolate or oblate ellipsoid with the axial ratio (a/b) of the ellipsoid ///o is the ratio of the frictional coefficient of a body (such as an ellipsoid) to that of a sphere of equal volume. For a sphere, a/b =1.) From J.T. Edward, Ref. [1 l,a].

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