Director fluctuations and nuclear-spin relaxation in lyotropic nematic liquid crystals

Lyotropic nematic liquid crystals, like their thermotropic counterparts, exhibit collective reorientation modes known as director fluctuations. In this work we consider the effect of director fluctuations on the transverse spin relaxation of quadrupolar nuclei in uniaxial lyotropic nematic liquid crystals, reporting ${}_{}{}^2{}_{}{}^{}\mathrm{H}$ (labeled surfactant) and ${}_{}{}^{23}{}_{}{}^{}\mathrm{Na}$ (counterion) relaxation data from the calamitic (${\mathit{N}}_{\mathit{C}}^{+}$) and discotic (${\mathit{N}}_{\mathit{D}}^{\mathrm{-}}$) phases of the system sodium dodecyl sulphate-decanol-water. The transverse relaxation is found to be dominated by a large contribution to the secular zero-frequency spectral density from diffusion-modulated (counterion) or viscoelastic (surfactant) director fluctuations. The existing theory of nuclear-spin relaxation by director fluctuations is extended to include the effect of translational diffusion on the fourth-order director-fluctuation time correlation functions. In contrast to thermotropic nematics, the second-order director-fluctuation contribution to the nonsecular high-frequency spectral densities is negligible in lyotropic nematic liquid crystals at conventional magnetic fields. This is a consequence of the much longer short-wavelength cutoff in lyotropic liquid crystals. The large zero-frequency spectral density, however, allows us to deduce the viscoelastic properties of the nematic phases, obtaining effective elastic constants of 0.3–1.0 pN in the investigated temperature range. The nematic order parameter is found to be significantly influenced by collective modes as well as by local micelle reorientation. For the oblate micelles in the ${\mathit{N}}_{\mathit{D}}^{\mathrm{-}}$ phase, the local modes are most important, whereas the reverse is true for the prolate micelles in the ${\mathit{N}}_{\mathit{C}}^{+}$ phase. In the surfactant case, the longest-wavelength-director-fluctuation modes are too slow to motionally average the nuclear-quadrupole coupling, resulting in a static broadening of the ${}_{}{}^2{}_{}{}^{}\mathrm{H}$ satellites.