Length scale for the onset of Fickian diffusion in super- cooled liquids

– The interplay between self-diffusion and excitation lines in space-time was recently studied in kinetically constrained models to explain the breakdown of the Stokes-Einstein law in supercooled liquids. Here, we further examine this interplay and its manifestation in incoherent scattering functions. In particular, we establish a dynamic length scale below which Fickian diffusion breaks down, as is observed in experiments and simulations. We describe the temperature dependence of this length scale in liquids of various fragilities, and provide analytical estimates for the van Hove and self-intermediate scattering functions. A ten-day journey starts with a single step. — Laotse, Tao Te King. In this paper, we consider the process of self-diffusion of probe molecules in supercooled liquids. Figure 1 shows the trajectory of a such a probe obtained from a model of a supercooled liquid [1]. At conditions shown, the structural relaxation time of the model is of the order of 10 microscopic time steps. The left panel of Fig. 1 extends over this range of time. The right panel extends three orders of magnitude longer in time, and one order of magnitude larger in space. Here, the trajectory looks like a random walk of Fickian diffusion, unlike the trajectory in the left panel. In this paper, we describe the crossover from non-Fickian to Fickian diffusion, and identify the length scale, l, that characterizes the crossover. Supercooled liquids can be studied theoretically using simple models where the density field dynamics is mapped to a coarse-grained mobility field evolving with simple empirical rules [2]. The main feature of these models is that their dynamics becomes spatially correlated [3], i.e., the growth of timescales is accompanied by the growth of dynamical lengthscales, giving rise to the phenomenon of dynamic heterogeneity observed in experiments and simulations [4–7]. Having at hand microscopic models capturing the essential fluctuations, it is important to revisit in detail all sorts of experimental and numerical studies of supercooled liquids within this coarse-grained approach [1,8–10]. Generic properties are only weakly dependent upon details of the models, which become important, however, for quantitative comparisons to experiments or simulations [8]. In most of this paper we will therefore pursue our investigations in the simplest lattice model of this family [2], namely the one-dimensional Fredrickson-Andersen model