  The multi-phase-field approach is generalized to treat capillarity-driven
diffusion parallel to the surfaces and phase-boundaries, i.e. the boundaries
between a condensed phase and its vapor and the boundaries between two or
multiple condensed phases. The effect of capillarity is modeled via
curvature-dependence of the chemical potential whose gradient gives rise to
diffusion. The model is used to study thermal grooving on the surface of a
polycrystalline body. Decaying oscillations of the surface profile during
thermal grooving, postulated by Hillert long ago but reported only in few
studies so far, are observed and discussed. Furthermore, annealing of
multi-nano-clusters on a deformable free surface is investigated using the
proposed model. Results of these simulations suggest that the characteristic
crater-like structure with an elevated perimeter, observed in recent
experiments, is a transient non-equilibrium state during the annealing process.
