Growth of a single drop formed by diffusion and adsorption of monomers on a two-dimensional substrate

We study a single, motionless three-dimensional (3D) drop growing by adsorption of diffusing monomers on a 2D substrate. A simple treatment based on a quasistatic approximation predicts that the radius of the drop increases as [t/ln(t)]1/3 in the long-time limit. By applying the method of matched asymptotic expansions we then confirm that the quasistatic approximation provides a dominant asymptotic behavior. We also show that the typical distance from the surface of the growing drop to the nearest surviving monomer scales as [ln(t)]1/2 and discuss the distribution function for that minimum distance.