The aim of this paper is to study continuum models for surface diffusion
taking into account free adatoms on the surface, which is of particular importance
in the (self-assembled) growth of nanostructures. The extended model yields
a coupled system of parabolic differential equations for the surface morphology and
the adatom density, involving a cross-diffusion structure.
¶ We investigate two different situations, namely the growth of a film on
a substrate and the growth of a crystal-like structure (a closed curve or surface).
An investigation of the equilibrium situation, which can be phrased as an energy minimization
problem subject to a mass constraint, shows a different behaviour in both
situations: for the film the equilibrium is attained when all atoms are
attached to the surface, while for a crystal the adatom density does not vanish
on the surface. The latter is also a deviation from the usual equilibrium theory,
since the equilibrium shape will be strictly included in the Wulff shape.
Moreover, it turns out that the total energy is not lower semicontinuous
and non-convex for large adatom densities and rough surfaces.
¶ The dynamics of the adatom surface diffusion model is investigated in detail
for situations close to a flat surface in the film case and the situation close
to a radially symmetric curve, both with an almost spatially homogeneous adatom density,
where the cross-diffusion structure of the model and the decay to equilibrium
can be studied in detail.
¶ Finally, we discuss the numerical solution of the adatom surface diffusion model
in the film case and provide various simulation results.