Rotary diffusion equation

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. 

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... 

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

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... 

In the case of rotary diffusion and frictional coefficients, only the rotation of the major a-axis about the minor 6-axis is in general experimentally measurable. Accordingly only the equations for 0(, and fi, will be considered here.)... 

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. 

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... 

Figure 11.23—Comparison of theo-retical and experimental first and second normal stress differences N and N2. The theoretical results (a) were calculated from the Smoluchowski equation (11-3) using the Onsager potential with U = 10.67, the minimum value for a fully nematic state, y/ >r is the dimensionless shear rate (or Deborah number), where Dr is the rotary diffusivity of a hypothetical isotropic fluid at the same concentration. Only the molecular-elastic contribution to the stress tensor was considered. The experimental results (b) are for 12.5% (by weight) PBLG (molecular weight = 238,000) in w-cresol. (Reprinted with permission from Magda et al., Macromolecules 24 4460. Copyright 1991, American Chemical Society.)...

For many purposes, it is more convenient to characterize the rotary Brownian movement by another quantity, the relaxation time t. We may imagine the molecules oriented by an external force so that the a axes are all parallel to the x axis (which is fixed in space). If this force is suddenly removed, the Brownian movement leads to their disorientation. The position of any molecule after an interval of time may be characterized by the cosine of the angle between its a axis and the x axis. (The molecule is now considered to be free to turn in any direction in space —its motion is not confined to a single plane, but instead may have components about both the b and c axes.) When the mean value of cosine for the entire system of molecules has fallen to ile(e — 2.718... is the base of natural logarithmus), the elapsed time is defined as the relaxation time r, for motion of the a axis. The relaxation time is greater, the greater the resistance of the medium to rotation of the molecule about this axis, and it is found that a simple reciprocal relation exists between the three relaxation times, Tj, for rotation of each of the axes, and the corresponding rotary diffusion constants defined in equation (i[Pg.138]

Generalizing the hydrod3mamical equations derived by Stokes for spheres, Edwardes (35) calculated the coefficients fj, Ca and for ellipsoids as a function of their axial ratios. The general equations are complicated but for ellipsoids of revolution, which may be characterized by only two values of f, they assume a simpler form, and have been employed by Gans 47) and F. Perrin 92) to evaluate the rotary diffusion constants of molecules which may be treated as ellipsoids of revolution. The formulas of Gans and Perrin are not identical, but the numerical values of 0 calculated from them are nearly so, so that the formulas of either author may be used in practice. In the following discussion we shall employ Perrin s equations. 

Consider first the case of an elongated ellipsoid of revolution a >b). Rotary Brownian movement of the a axis about the b axis is characterized by the relaxation time and the corresponding rotary diffusion constant 0 = 1/2t [see Equation (19a)]. These constants are conveniently expressed by their values relative to those for a sphere of the same volume. Denoting by q the ratio bja, Perrin s equation reads... 

If the rotating cylinder could be brought suddenly and smoothly to rest, the second term on the right would vanish, and the resulting equation would describe the process of disorientation, by free rotary diffusion, of the molecules which had previously been partly oriented by the flow. The speed of disorientation is proportional to the rotary diffusion constant, 0, and the final steady state of random distribution corresponds to the equation q = const. = Nj4 n. 

If the values of co and co from (33) and (34) are substituted in (32), a differential equation for the function q is obtained, for which a general solution by expansion in an infinite series has been obtained by Peterlin 96), 98), 99). The solution is best expressed with the aid of the parameter a = GjB, the ratio of velocity gradient to rotary diffusion constant. The general solution is very complex, and the terms converge slowly but for low values of [Pg.146]

For prolate ellipsoids, the rotary diffusion coefficient D, is given by Perrin equation (IS) ... 

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) ... 

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