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**Surface diffusion including adatoms.**
*(English)*
Zbl 1152.74379

Summary: 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-diffion 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.

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.

### MSC:

74N20 | Dynamics of phase boundaries in solids |

74N25 | Transformations involving diffusion in solids |

35K50 | Systems of parabolic equations, boundary value problems (MSC2000) |

35K55 | Nonlinear parabolic equations |

65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |