Rotary diffusion coefficient

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. 

From here the rotary diffusion coefficient is expressed as... 

For solutions of nonspherical particles the situation is more complicated and the physical picture can be described qualitatively as follows for a system of particles in a fluid one can define a distribution function, F (Peterlin, 1938), which specifies the relative number of particles with their axes pointed in a particular direction. Under the influence of an applied shearing stress a gradient of the distribution function, dFfdt, is set up and the particles tend to rotate at rates which depend upon their orientation, so that they remain longer with their major axes in position parallel to the flow than perpendicular to it. This preferred orientation is however opposed by the rotary Brownian motion of the particles which tends to level out the distribution or orientations and lead the particles back toward a more random distribution. The intensity of the Brownian motion can be characterized by a rotary diffusion coefficient 0. Mathematically one can write for a laminar, steady-state flow ... 

According to Perrin (1934), the rotary frictional coefficient, f, and thereby the rotary diffusion coefficient, 0, for ellipsoids of revolution can be given as... 

Thus, like / the rotary diffusion coefficient, f, also depends on both the volume and the axial ratio of the particles. 

To avoid possible confusion with the symbol for rotary diffusion coefficient here we have used the symbol 0 for the theta temperature, rather than 6 which was originally used by Flory. 

As mentioned earlier, ideally the best answer should come from the determination of the 5-function. Unfortunately at present the rotary diffusion coefficient is usually the least reliable quantity in all hydrodynamic measurements because of errors inherent in the physical methods of flow birefringence and perhaps also non-Newtonian viscosity (see Section IV). (Electric birefringence also may not give the same rotary diffusion coefficient as the other two methods, since the equivalent ellipsoids can be different under shearing stress and under electrical field.) Edsall (1954) has also illustrated the impossibility of evaluating the axial ratio from the 5-function. The latter was about 0.80 for fibrinogen which corresponded to a prolate ellipsoid with an axial ratio of more than 300. If the rotary diffusion coefficient were only about 15% greater than that listed in Table V the calculated axial ratio would decrease to between ten and twenty. 

Since the gradient dependence of the intrinsic viscosity of rigid particles gives a direct measure of the rotary diffusion coefficient it is po.ssible in... 

With the development of the non-Newtonian viscosity theories it is now possible to compare the rotary diffusion coefficient and thereby the calculated length (or diameter) of the rigid particles as obtained from this technique with that from the commonly used flow birefringence method. Since both measurements depend upon the same molecular distribution function (Section III) they should give an identical measure of the rotary diffusion coefficient. Differences, however, will arise if the system under study is heterogeneous. The mean intrinsic viscosity is calculated from Eq. (7) whereas the mean extinction angle, x, for flow birefringence is defined by the Sadron equation (1938) ... 

For ellipsoids of revolution the numerical values of va and vb have been tabulated by Scheraga (1955), and the sum of va and vb (i.e., vr at oj = 0) is identical with the viscosity increment from Simha s equation. Thus Eq. (43) provides an alternative method to that of the non-Newtonian viscosity for the determination of the rotary diffusion coefficient, 0. Cerf has also pointed out that 0 is determinable from the slope at the inflection point (I.P.) of the vr versus w-curve, i.e., w(I.P.) = 2 /30. At present, however, no experimental test of Eq. (43) has as yet been reported. 

For prolate ellipsoid, the rotary diffusion coefficient Dt ( >, = l/6r) is given by the Perrin formula [47]... 

One of the important applications of Stokes law occurs in the theory of Brownian motion. According to Einstein (El a) the translational and rotary diffusion coefficients for a spherical particle of radius a diffusing in a medium of viscosity n are, respectively,... 

If Dg is the rotary diffusion coefficient of collagen molecule, we can introduce three different critical times ... 

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