Deuterium NMR relaxation-time constants T1Q and T1Z from spectra of a phenyl-ring-deuterated double-tailed surfactant, sodium 4-(1-heptylnonyl)benzenesulfonate (SHBS), in liquid crystals formed with water were obtained from 0 to 80-degrees-C. The relaxation-time constants were determined for the 15-degrees and 90-degrees orientations of the powder sample. These data are compared to predicted relaxation-time constants from two different models for molecular reorientation, including (I) phenyl rings undergoing uniaxial reorientation about the 1’-4’ axis and (II) phenyl rings rotating rapidly about the 1’-4’ axis of a molecule which is reorienting about a space-fixed director axis. Both models include an adjustable parameter that describes molecular motion about the 1’-4’ axis that results from either strong collisions or small step free rotational diffusion. The results from both models indicate the rate of reorientation about the 1’-4’ axis of the phenyl ring is approximately 3 times the nuclear Larmor frequency ofthe deuterium nucleus, 30 MHz, at room temperature. This rate increases with increasing temperature. The best fits of the data were determined using model II. The predicted rate constants for motion about this axis differ slightly for each model, whereas the type of motion predicted by the two models is very different. Model I indicates reorientation about the 1’-4’ axis is more like strong collisions. Model II indicates that the motion is more like small step rotational diffusion.