NMR Relaxation via Translational Diffusion

The direct NMR method for determining translational difiFusion constants in liquid crystals was described previously. The indirect NMR methods involve measurements of spin-lattice relaxation times (Ti,Ti ),Tip) [7.45]. Prom their temperature and frequency dependences, it is hoped to gain information on the self-diflPusion. In favorable cases, where detailed theories of spin relaxation exist, difiFusion constants may be calculated. Such theories, in principle, can be developed [7.16] for translational difiFusion. However, until recently, only a relaxation theory of translational difiFusion in isotropic hquids or cubic solids was available [7.66-7.68]. This has been used to obtain the difiFusion correlation times in nematic and smectic phases [7.69-7.71]. Further, an average translational difiFusion constant may be estimated if the mean square displacement is known. However, accurate determination of the difiFusion correlation times is possible in liquid crystals provided that a proper theory of translational difiFusion is available for liquid crystals, and the contribution of this difiFusion to the overall relaxation rate is known. In practice, all of the other relaxation mechanisms must first be identified and their contributions subtracted from the observed spin relaxation rate so as to isolate the contribution from translational difiFusion. This often requires careful measurements of proton Ti over a very wide frequency range [7.72]. For spin - nuclei, dipolar interactions may be modulated by intramolecular (e.g., collective motion, reorientation) and/or intermolecular (e.g., self-diffusion) processes. Because the intramolecular (Ti ) and intermolecular 

FIGURE 7.6. Plots of self-diffusion constants versus the reciprocal temperature in three discotic liquid crystals. The circles and squares correspond to diffusion along z and x, respectively, while the triangles refer to the isotropic liquid. The open and filled symbols refer to different experimental runs. The vertical bars represent the uncertainty in the result for the single measurements (after Ref. [7.65]). 

The theory of Torrey [7.66, 7.67, 7.75] treats relaxation via dipolar interactions between spins on different molecules in an isotropic liquid. An extension of this relaxation theory to the case of translational diffusion in nematic [7.76], smectic A [7.77], and smectic B phases [7.78] has recently been developed. Zumer and Vilfan assumed an elongated cylindrical shape for the molecule with a particular distribution of spins on the cylindrical surface and a perfect ordered ((P2) = 1) system. Their expression for Ti due to self-diffusion in nematics is given by [7.76] 

Nordio, The Molecular Physics of Liquid Crystals, edited by G.R. Luckhurst and G.W. Gray (Academic Press, London, 1979), Chap. 18. 

Petersen, Electron Spin Relaxation in Liquids, edited by L.T. Muus and P.W. Atkins (Plenum Press, New York, 1972). 

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