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Apart from three-dimensional continuum and discrete models, the evolution of surfaces is usually described by spatially two-dimensional partial differential equations (PDEs). These models are often derived from or at least motivated by small gradient approximations, but the studied surfaces do not fulfill this requirement in all cases. We will investigate how to overcome the small gradient approximation by using geometric PDEs. Therefore, we will introduce a method to simulate the evolution of surfaces with respect to local geometric properties. In contrast to traditional PDEs, this method does not depend on the parametrization of the surface. It will not only allow us to simulate surface evolution on flat geometries but also on more complex shaped objects. For small gradients, the studies of simple model equations show similar results compared to the related PDEs. For large gradients the results differ fundamentally. Hence, the small gradient approximation should only be applied in cases where large gradients does not appear. Specifically, we exemplify this using various equations including the (damped) Kuramoto-Sivashinsky equation, which is used as a minimal model for low-energetic erosion and deposition processes, and its geometric PDE counterpart. [ABSTRACT FROM AUTHOR]

Using a continuum formulation valid for arbitrary laterally two-dimensional surface profiles, we investigate the significance of redeposition of ion-beam eroded particles on spatio-temporally evolving surface morphologies of the target. Combining the effect of redeposition with standard roughening and smoothing effects of ion-beam erosion processes to a generalized nonlocal Kuramoto-Sivashinsky equation, we substantiate the decisive role of redeposition for the emergence of well-ordered ion-beam eroded hexagonally arranged dot structures as, e.g., experimentally observed on various semiconductor targets. [ABSTRACT FROM AUTHOR]

A nonlinear stochastic growth equation for the spatiotemporal evolution of the surface morphology of amorphous thin films in the presence of potential density variations is derived from the relevant physical symmetries and compared to recent experimental results. Numerical simulations of the growth equation exhibit a saturation of the surface morphology for large film thickness originating from the inclusion of the density inhomogeneities. Furthermore, we argue why moundlike surface structures observed on vapor deposited amorphous films are not the result of the Grinfeld instability.

Granular systems can compact under the influence of sufficiently strong, successive tapping. Recent experimental investigations show that the packing fraction obeys a very slow relaxation to a final, dense packing fraction that is basically proportional to the inverse of the logarithm of the tap number or time. We provide a simple macromechanical argument that explains this inverse logarithmic relaxation in time in all functional details. By considering the asymptotic limits of the resulting relaxation law, we show that the relaxational dynamics of the compaction process can be interpreted as a combination of a biased void diffusion for short times and a collective reorganization for large times.