Crossover between 2D and 3D Fluid Dynamics in the Diffusion of Islands in Ultrathin Freely Suspended Smectic Films

The Stokes paradox, that moving a disk at finite velocity through an infinite two-dimensional (2D) viscous fluid requires no force, leads, via the Einstein relation, to an infinite diffusion coefficient D for the disk. Saffman and Delbruck proposed that if the 2D fluid is a thin film immersed in a 3D viscous medium, then the film should behave as if it were of finite size, and D similar to - In(a eta'), where a is the inclusion radius and eta' is the viscosity of the 3D medium. By studying the Brownian motion of islands in freely suspended smectic liquid crystal films a few molecular layers thick, we verify this dependence using no free parameters, and confirm the subsequent prediction by Hughes, Pailthorpe, and White of a crossover to 3D Stokes-like behavior when the diffusing island is sufficiently large.