The Rotational Diffusion Model

We now use the Debye equation to treat molecular reorientation in a nematic liquid crystal [7.1, 7.2] by suitably modifying the equation to allow for an ordering potential that is known to be present in the mesophase. The orientational distribution function f Cl) and the conditional probability function f Qo n,t) are obtained by solving 

In the absence of the ordering potential, the eigenfunctions of both Fq and Tq are the Wigner functions (f]), since 

Here for the moment tum is used instead of n to emphasize that n is a projection index onto the molecular. 2 axis, and the eigenvalues otL mM 

For cylindrical molecules in mesophases of Dooh symmetry, a truncated form of the pseudo-potential (Maier-Saupe potential) may be used, 

In this special case, Tq does not connect Wigner functions with diflPerent m or n indices, and so the eigenfunctions of Tq are given by 

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Models for description of liquids should provide us with an understanding of the dynamic behavior of the molecules, and thus of the routes of chemical reactions in the liquids. While it is often relatively easy to describe the molecular structure and dynamics of the gaseous or the solid state, this is not true for the liquid state. Molecules in liquids can perform vibrations, rotations, and translations. A successful model often used for the description of molecular rotational processes in liquids is the rotational diffusion model, in which it is assumed that the molecules rotate by small angular steps about the molecular rotation axes. One quantity to describe the rotational speed of molecules is the reorientational correlation time T, which is a measure for the average time elapsed when a molecule has rotated through an angle of the order of 1 radian, or approximately 60°. It is indirectly proportional to the velocity of rotational motion. 

The rotational diffusion model was generalized by Gordon (1966) to include... 

The correlation time, in Eq. (4) is generally used in the rotational diffusion model of a liquid, which is concerned with the reorientational motion of a molecule as being impelled by a viscosity-related frictional force (Stokes-Einstein-Debye model). Gierer and Wirtz have introduced the idea of a micro viscosity, The reorientational... 

The basic concepts of linear response theory are best illustrated by considering the rotational diffusion model of an assembly of electric dipoles constrained to rotate in two dimensions due to Debye [14] which is governed by the Smoluchowski equation... 

In the following sections we evaluate the orientational correlation functions [Eq. (7.2.6)] for the rotational diffusion model. 

R. J. Hall and D. A. Greenhalgh, Application of the Rotational Diffusion Model to Gaseous N2 CARS Spectra, Opt. Comm. 40 417 (1982). 

Use of the G.L.E. can lead to other refinements to the rotational diffusion model for example, Keyes (10) has noted that mutual rotational diffusion, where the relative reorientation of pairs of molecules is analyzed, can also be modelled in this way, with results obtainable not only for the simple linear diffusers treated earlier by Keyes and Kivelson (8), but for the as3mimetric diffuser as well. 

Note that the lower index of the D-function is a measure of the importance of rotation around the symmetry axis in affecting these band shapes, since the functions contain factors of cos[m(60f + 6Y)] To obtain some idea of the behavior of these functions, one might start with the rotational diffusion model which predicts that (16)... 

Small molecules such as perdeuterated TEMPONE (PDT) and 3-doxylcholes-tane (CSL) were the first nitroxides for which rotational dynamics were characterized in detail by multifrequency EPR at 9 and 250 Onz. " These studies permitted an initial exploration of the sensitivity of HF EPR to anisotropy of rotational diffusion and also allowed Freed and co-workers to test diffusion models critically. In order to explain the multifrequency EPR data, the rotational diffusion model has had to be extended from an axial approximation to the fully anisotropic case. This required some modifications of the existing fast-... 

Experimental measurements of dielectric relaxation confirm qualitatively the predictions of the rotational diffusion model, in that g has a low and high frequency relaxation, while g only shows relaxations at higher frequencies. Unfortunately there are few liquid crystal systems that have been studied over wide frequency ranges, and measurements at high frequencies >50 MHz on aligned samples are difficult. Some typical results are shown in Fig. 19. 

Deuteron NMR spectroscopy has been widely employed [2-6] to determine experimental values for the diffusion tensor elements. The coimection of NMR spectroscopy to the rotational diffusion model comes through the pertinent correlation fiinctions. In fact, the orientational correlation functions for a tensor of rank L are written, in general, as... 

The (3 and a coefficients in EQN (4) and the elements of the rotational diffusion tensor in EQN (5) are the results of the fitting procedure of the NMR data sketched previously, and their values depend on the details of the rotational diffusion model adopted. Expressions (5) are often written in terms of xx and X relaxation times, namely... 

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