Stochastic modelling of roto-translational motion of dyes in micellar environment

Time decay of the fluorescence anisotropy ratio due to diffusion of dyes in micelles is usually interpreted by decoupling the reorientational dynamics of the molecule with respect to the local director from the translational diffusion of the dye. However, while such a kind of assumption is justified in other contexts (like for reorientations of a small mobile fragment in a macromolecule, decoupled from the motion of the macromolecule as a whole, as invoked in the well-known model by Lipari and Szabo), here it is not based on physical grounds. In this work we develop the stochastic model for the full description of the roto-translational dynamics of a dye in the micellar environment, by employing the Fokker-Planck-Smoluchowski equation for the positional and orientational variables. Then we simplify the model to the situation of strong confinement of the molecule at the micelle interface. Finally, by employing a time-scale separation between fast reorientational dynamics and slow lateral diffusion of the dye (which holds if the micelle radius is much larger than the size of the dye), and by resorting to a model like the "wobbling in a cone", we show that a bi-exponential form can be obtained for the fluorescence anisotropy ratio, but with the remarkable difference that the fast-relaxing component is not affected by the slow motion.