Nuclear spin relaxation and molecular dynamics in ordered systems: Models for molecular reorientation in thermotropic liquid crystals

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Abstract
New models are developed to account for rotational motion in liquid crystals. A distinction is made between rotation about a molecule fixed z axis, described by Eulerian angle γ, and about a space fixed z axis, described by Eulerian angle α. Our model allows γ motion to proceed by jumps of arbitrary angular size, while motion about the space fixed axes (α,β motion) is described in terms of small step rotational diffusion in presence of a restoring pseudopotential. Calculations are presented for different forms of the restoring potential including (1−cosn β) for n=2, 4, and 10 where β is the angle between the molecule fixed and space fixed z axes, as well as angular square well potentials with finite and infinite walls. Multiexponential correction functions for α,β motion in the infinite square well (‘‘diffusion-in-a-cone’’) potential as well as the Maier–Saupe potential (cos2 β) are listed in tabular form as a function of the second rank order parameter Szz =〈 P2(cos β)〉. It is shown that spectral density functions relevant for calculation of 2H spin–lattice relaxation behavior are not very sensitive to the form of the restoring potential. J00(0), which contributes primarily to T2, appears to be more sensitive to the shape of the potential. Experimental spectral density ratios J1/J2 for numerous solutes and liquid crystalline solvent molecules can readily be explained in terms of dominant contributions from γ motion. In general, this motion seems to proceed by jumps of larger angular size as the degree of order increases.