A random walk description of mobility in glasses

Molecular motion is believed to be the key to the glass transition and the glassy behavior in general. Yet in the theoretical models the time dependent quantities are continuous e.g., density modes. Particles in condensed matter interact strongly and there does not seem to be a method of describing their movement short of the MD simulation. Here we investigate if a single particle model of molecular motion can capture at least some characteristic features of the glass forming systems. The target properties are the mean squared displacement (MSD) and the orientational autocorrelators as simulated via MD or extracted from the spectroscopic data. These properties exhibit rich behaviors in liquid and glassy state; e.g., the MSD exhibits a sub-diffusive plateau followed at longer times by asymptotic diffusive regime, where the length of the plateau increases dramatically with cooling toward the glass transition.

The translational diffusive behavior as well as the Debye rotational relaxation can be modeled as a discrete random walk. But what about the other features, including the MSD plateau? We analyze a sequence of random walk models of increasing complexity to establish what is required to reproduce the experimentally observed features of translational and rotational relaxation. The models include biased random walk and random walk on a cage structure of different topology in one, two, and three dimensions. The random walk models are able to account for: the sub-diffusive regime, dynamic heterogeneity, emergence of the α- and β- relaxation processes, wedge-like shape of the α- relaxation spectrum, and the relative rate of decay of the first and second Legendre polynomials of the cosine of the rotation angle.  Unlike the traditional models, the random walk models do not involve overcoming potential energy barriers. The characteristic slowing down of the mobility in the random walk-based models results from increasing tortuosity of the path a molecule follows.