Relaxation dynamics of a linear molecule in a random static medium: A scaling analysis

We present extensive molecular dynamics simulations of the motion of a single linear rigid molecule in a two-dimensional random array of fixed obstacles. The diffusion constant for the center of mass translation, $D_{\rm CM}$, and for rotation, $D_{\rm R}$, are calculated for a wide range of the molecular length, $L$, and the density of obstacles, $\rho$. The obtained results follow a master curve $D\rho^{\mu} \sim (L^{2}\rho)^{-\nu}$ with an exponent $\mu = -3/4$ and 1/4 for $D_{\rm R}$ and $D_{\rm CM}$ respectively, that can be deduced from simple scaling and kinematic arguments. The non-trivial positive exponent $\nu$ shows an abrupt crossover at $L^{2}\rho = \zeta_{1}$. For $D_{\rm CM}$ we find a second crossover at $L^{2}\rho = \zeta_{2}$. The values of $\zeta_{1}$ and $\zeta_{2}$ correspond to the average minor and major axis of the elliptic holes that characterize the random configuration of the obstacles. A violation of the Stokes-Einstein-Debye relation is observed for $L^{2}\rho > \zeta_{1}$, in analogy with the phenomenon of enhanced translational diffusion observed in supercooled liquids close to the glass transition temperature.