Diffusivity, rotary orientation dependence

The difference between Eq. (9.1) and the analogous equation for flexible homopolymers (Eq. (4.69)) is in the orientation-dependent rotary diffusivity D. which depends on the anisotropy of the system, as expressed by (Doi and Edwards 1978a, 1978b) ... 

Returning to the solution of Eq. (9.1), for simplification Doi and Edwards (1978b) replaced the orientation-dependent rotary diffusivity (see Eq. (9.4)) with an averaged orientation-independent quantity D ... 

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. 

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 ... 

First, all molecules or particles rotate in a shear flow, but if they are not precisely spherical, the flow causes an orientation aligned in the direction of the flow to last for a longer time than orientation perpendicular to the flow (Section 5.1.1). This implies that average flow disturbance, and thereby viscosity, is smaller. The alignment depends on the rate of rotary diffusion of the particles in relation to the magnitude of the shear rate. The rotary... 

---