Director Fluctuations and Spin Relaxation

In the preceding chapter, it was found that the magnitudes of the motional frequency components at the Larmor frequency, at twice this frequency, and at zero frequency are important in NMR relaxation. Both spin-lattice and spin-spin relaxation rates are generally governed by random motion of spin-bearing molecules. Typically, a molecule remains in one state of motion for a short time s). After this time, it suffers a col- 

Collective motions are elastic deformations in liquid crystalline samples, but can also exist in their isotropic phases with a finite coherence length, giving rise to pretransitional phenomena [6.2]. These motions are perceived as hydrodynamic phenomena and are influenced by molecular properties such as elastic constants and viscosities of the liquid crystalline medium. At best, director fluctuations can only provide indirect information on the anisotropic intermolecular interactions. On the contrary, motion on a molecular level must reflect the shape of the instantaneous potential of mean torque on each molecule. Both both molecular rotation and translation are expected to be sensitive to the nature of anisotropic interactions, which determine the formation of various liquid crystalline structures. 

The extension of a molecular dynamic model for simple liquids to anisotropic mesophases will be deferred to the next chapter. In Section 6.1, spin 

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In-depth treatments of the topic are available in several books [1-6] and in a large number of review articles. The monograph by Dong [6], for example, focuses on aspects like the dynamics of nuclear spins, orientational order, molecular field theories, nuclear spin relaxation theory, director fluctuation and spin relaxation, rotational and translational dynamics, internal dynamics of flexible mesogens, and multiple-quantum and two-dimensional NMR, topics that will be touched upon very briefly here. Re-... 

In contrast to nematics, a helical twist of the molecular director takes place in the chiral nematic phase. Studies of the spin-lattice relaxation in chiral nematics have shown that the relaxation mechanisms are essentially the same as in pure nematics [141, 142]. At high Larmor frequencies the relaxation is diminished by molecular self-diffusion and by local molecular rotations, whereas director fluctuations determine the relaxation rate at low Larmor frequencies. This can be easily understood because the spin-lattice relaxation rate in the MHz region is dominated by orientational fluctuations with wavelength much smaller than the period of the helix. The influence upon the rotating frame spin-lattice relaxation time Tip of the rotation of the molecules due to diffusion along the helix, an effect specific for twisted structures, has not been observed in COC [143]. 

In the nematic phase the spin-lattice relaxation rate at high Larmor frequencies is determined by local reorientations of the molecule and internal molecular motions. The spin-spin relaxation rate, I/T2, on the other hand, is determined by nematic order director fluctuations and rotations induced by translational diffusion [216]. 

Proton, deuteron and carbon spin relaxation measurements of liquid crystals have provided detailed information about the molecular motions of such anisotropic liquids (anisotropic rotation and translation diffusion of individual molecules), and about a peculiar feature of liquid crystalline phases, namely collective molecular reorientations or order fluctuations. Spin relaxation in liquid crystalline mesophases has challenged NMR groups since the early 1970s, shortly after the publication of theoretical predictions that order fluctuations of the director (OFD, OF), i.e. thermal excitations of the long-range orientational molecular alignment (director), may play an important unusual role in nuclear spin relaxation of ordered liquids. Unique to these materials, which are composed of rod-like or disc-like (i.e. strongly anisotropic molecules), it was predicted that such thermal fluctuations of the director should, at the frequencies of these fluctuation modes, produce rather peculiar Ti(p) dispersion profiles. For example in the case of uniaxial nematic... 

It is considerably larger in the confined liquid crystals above Tni than in the bulk isotropic phase. The additional relaxation mechanism is obviously related to molecular dynamics in the kHz or low MHz frequency range. This mechanism could be either order fluctuations, which produce the well-known low-frequency relaxation mechanism in the bulk nematic phase [3], or molecular translational diffusion. Ziherl and Zumer demonstrated that order fluctuations in the boundary layer, which could provide a contribution to are fluctuations in the thickness of the layer and director fluctuations within the layer [36]. However, these modes differ from the fluctuations in the bulk isotropic phase only in a narrow temperatnre range of about IK above Tni, and are in general not localized except in the case of complete wetting of the substrate by the nematic phase. As the experimental data show a strong deviation of T2 from the bulk values over a broad temperature interval of at least 15K (Fig. 2.12), the second candidate, i.e. molecular translational diffusion, should be responsible for the faster spin relaxation at low frequencies in the confined state. 

This chapter concludes by pointing out that relaxation of multispin proton systems played a major role in the early days of NMR relaxation measurements on liquid crystals [5.34]. In particular, the detection of director fluctuations [5.35] by means of its characteristic frequency dependence in proton Ti [5.36-5.39] started intensive NMR research on liquid crystals. Since there are many inequivalent proton species in a liquid crystalline molecule, it is impossible to distinguish various atomic sites from a broad proton lineshape, which is a result of strong dipolar couplings. Moreover, translation self-diffusion also modulates the intermolecular dipole-dipole interactions and contributes to proton relaxation in liquid crystals [5.40, 5.41]. Partially deuterated liquid crystals may be used to reduce the number of inequivalent proton species. Proton spin relaxation studies remain as a possible method of probing intermolecular interactions or translational motions in liquid crystals. 

There are several motional processes that take place simultaneously and may cause spin relaxation in liquid crystals. To incorporate the time scales and amplitudes of various physical motions in liquid crystals, it is necessary to consider different coordinate systems. Because of thermal fluctuations of the director, the orientation of the director has both spatial and temporal variations. A local (or instantaneous) director n(r ) may be defined to represent the direction of preferred orientation of the molecules in the neighborhood of any point in the sample. Thus, an additional coordinate system is needed to specify the local director n(r ). The average director no is obtained by spatially averaging the local directors over the sample at any particular instant. Now the motional spectral densities are given by Eq. (5.31) ... 

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