The crossover from single file to Fickian diffusion

The crossover from single-file diffusion, where the mean-square displacement scales as <x^2> ~t^(1/2), to normal Fickian diffusion, where <x^2>~t$, is studied as a function of channel width for colloidal particles. By comparing Brownian dynamics to a hybrid molecular dynamics and mesoscopic simulation technique, we can study the effect of hydrodynamic interactions on the single file mobility and on the crossover to Fickian diffusion for wider channel widths. For disc-like particles with a steep interparticle repulsion, the single file mobilities for different particle densities are well described by the exactly solvable hard-rod model. This holds both for simulations that include hydrodynamics, as well as for those that don't. When the single file constraint is lifted, then for particles of diameter \sigma and pipe of width$L$ such that (L- 2 \sigma)/\sigma = \delta_c << 1 the particles can be described as hopping past one-another in an average time t_{hop}. For shorter times t << t_{hop} the particles still exhibit sub-diffusive behaviour, but at longer times t > t_{hop}, normal Fickian diffusion sets in with an effective diffusion constant D_{hop} ~ t_{hop}^(1/2). For the Brownian particles, t_{hop} ~ 1/\delta_c^(2) when \delta << 1, but when hydrodynamic interactions are included, we find a stronger dependence than \delta_c^{-2}. We attribute this difference to short-range lubrication forces that make it more difficult for particles to hop past each other in very narrow channels.