A theory is presented which describes the capillary-driven aging of discontinuous thin films on a substrate, where the primary transport mechanism among the domains is two-dimensional diffusion of species over the substrate. This theory employs a statistical dynamics formulation, whereby the average growth rate for each domain size class is determined relative to the critical (zero-growth) domain size. The time dependence of the critical size is determined through a global constraint on the individual fields. The effect of fractional area coverage, Aa, is accounted for through a second global constraint over the distribution of island sizes.
This theory yields a self-similar size distribution that is fairly insensitive to Aa. The critical island radius, R*, is found to increase asymptotically as the cube-root of time. The growth rate of R* increases with Aa, which results from the closer proximity of the islands and steeper concentration gradients as Aa increases.