Rotary Diffusivities

Brownian motion and interparticle interactions can produce deviations from this Jeffery orbit. When particle rotations are disturbed by Brownian motion, the orbits become stochastic 

For a particle of nearly spherical shape and diameter d we have 

For rods of length L and diameter d (Doi and Edwards 1986 Kirkwood and Auer 1951), DrQ is estimated by a formula similar to Eq. (6-32a)  

Equation (6-32b) is in good agreement with measured in dilute solutions of PBLG (Ookubo et al. 1976 Warren et al. 1973). A slightly more rigorous equation for Dro is given below, in Eqs. (6-37) and (6-38). Rotary diffusivities for spheroids and other shapes are 

The crossover from Brownian to non-Brownian behavior in a flowing suspension is controlled by a rotational Peclet number. 

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As with spherical particles the Peclet number is of great importance in describing the transitions in rheological behaviour. In order for the applied flow field to overcome the diffusive motion and shear thinning to be observed a Peclet number exceeding unity is required. However, we can define both rotational and translational Peclet numbers, depending upon which of the diffusive modes we consider most important to the flow we initiate. The most rapid diffusion is the rotational component and it is this that must be overcome in order to initiate flow. We can define this in terms of a diffusive timescale relative to the applied shear rate. The characteristic Maxwell time for rotary diffusion is... 

This means that for such particles the contribution by rotary diffusion is predominant but not exclusive. In fact, as a consequence of its rigidity, a rigid rod can only follow the rotary component of laminar shear flow There remains some radial flow of solvent along the particle which causes the hydrodynamic contribution to the intrinsic viscosity. 

For the calculation of the Maxwell-constant an assembly of frozen random conformations is considered. Brownian motion is taken into account only so far as rotary diffusion of the rigid conformations is concerned. In this way a first order approximation of the distribution function with respect to shear rate is obtained. This distribution function is used for the calculation of the Maxwell-constant, [cf. the calculation of the Maxwell-constant of an assembly of frozen dumb-bell models, as sketched in Section 5.I.3., eq. (5.22)]. Intrinsic viscosity is calculated for the same free-draining model, using average dimensions [cf. also Peter-lin (101)]. As for the initial deviation of the extinction angle curve from 45° a second order approximation of the distribution function is required, no extinction angles are given. 

Ellipsoidal or rod-shaped molecules have two different rotary diffusion constants while, if the dimensions of the molecules are different along all three axes, three constants must be specified.36... 

Rocky Mountain spotted fever 7 Rods (visual receptor cells) 390 Root hairs, dimensions of 30 Roseoflavin 788, 789s Rossmann fold. See Nucleotide-binding domain Rotamases 488 Rotary diffusion constant 463 Rotation of molecules 462,463 Rotational barrier 44 Rotifers 24, 25... 

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. 

The mathematical basis for our work is formed by the technique of solving the orientational rotary diffusion (Fokker-Planck) equation by reducing this... 

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

Note the close resemblance between the reference time of the internal rotary diffusion of the magnetic moment and the Debye time... 

The rotary diffusion (Fokker-Planck) equation for the distribution function W(e,t) of the unit vector of the particle magnetic moment was derived by Brown [47]. As shown in other studies [48,54], it may be reduced to a compact form... 

The kinetic (rotary diffusion) equation for the particle magnetic moment may be solved with high precision, thus taking into account contributions from the intrawell motions that are essential for a correct description of SR, especially at low temperatures. 

These processes have different timescales. For magnetic moments it is the time of internal superparamagnetic diffusion xD [see Eq. (4.28)], and for the axes alignment it is the time of mechanical rotary diffusion Tb of a particle in a carrier liquid [see Eq. (4.29)]. As once noticed in Ref. 48 (see also Section II.A above), both parameters may be presented in a similar form... 

The orientational kinetics of the dipolar suspension is described by the rotary diffusion equation presented in Section II as Eq. (4.51) its form for the electric dipoles is also well known [147,148], The only modification one has to perform in Eq. (4.51) to make it account for the particles suspended in a fluid is to change the relaxation time from xD to xB. Defining the latter in a more general form than in Eq. (4.29), we write... 

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