Frank Haußer

A numerical scheme for regularized anisotropic curve shortening flow

Abstract Realistic interfacial energy densities are often non-convex, which results in backward parabolic behavior of the corresponding anisotropic curve shortening flow, thereby inducing phenomena such as the formation of corners and facets. Adding a term that is quadratic in the curvature to the interfacial energy yields a regularized evolution equation for the interface, which is fourth-order parabolic. Using a semi-implicit time discretization, we present a variational formulation of this equation, which allows the use of linear finite elements. The resulting linear system is shown to be uniquely solvable. We also present numerical examples.

On Level Set Formulations for Anisotropic Mean Curvature Flow and Surface Diffusion

Anisotropic mean curvature motion and in particular anisotropic surface diffusion play a crucial role in the evolution of material interfaces. This evolution interacts with conservations laws in the adjacent phases on both sides of the interface and are frequently expected to undergo topological chances. Thus, a level set formulation is an appropriate way to describe the propagation. Here we recall a general approach for the integration of geometric gradient flows over level set ensembles and apply it to derive a variational formulation for the level set representation of anisotropic mean curvature motion and anisotropic surface flow. The variational formulation leads to a semi-implicit discretization and enables the use of linear finite elements.

A Step‐Flow Model for the Heteroepitaxial Growth of Strained, Substitutional, Binary Alloy Films with Phase Segregation: I. Theory

We develop a step‐flow model for the heteroepitaxy of a generic, strained, substitutional, binary alloy. The underlying theory is based on the fundamental principles of modern continuum thermodynamics. In order to resolve the inherent disparity in the spatial scales—continuous in the lateral directions vs. atomistically discrete along the epitaxial axis—we represent the film as a layered structure, with the layer height equal to the lattice parameter along the growth direction, thus extending the classical BCF framework [W. K. Burton, N. Cabrera, and F. C. Frank, Philos. Trans. Roy. Soc. London Ser. A, 243 (1951), pp. 299–358] to growth situations in which the bulk behavior impacts the surface evolution. Our discrete‐continuum model takes the form of a free‐boundary problem for the evolution of monoatomic steps on a vicinal surface, in which interfacial effects on the terraces and along the step edges couple to their bulk counterparts (i.e., within both film and, indirectly, substrate). In particular, the...

Return radius and volume of recrystallized material in Ostwald ripening.

Within the framework of the Lifshitz-Slyozov-Wagner theory of Ostwald ripening, the amount of volume of the second (solid) phase in a liquid solution that is newly formed by recrystallization is investigated. It is shown that in the late stage, the portion of the newly generated volume formed within an interval from time t(0) to t is a certain function of t/t(0) and an explicit expression of this volume is given. To achieve this, we introduce the notion of the return radius r(t,t(0)), which is the unique radius of a particle at time t(0) such that this particle has-after growing and shrinking-the same radius at time t. We derive a formula for the return radius, which later on is used to obtain the newly formed volume. Moreover, formulas for the growth rate of the return radius and the recrystallized material at time t(0) are derived.

Control of Nanostructures through Electric Fields and Related Free Boundary Problems

Geometric evolution equations, such as mean curvature flow and surface diffusion, play an important role in mathematical modeling in various fields, ranging from materials to life science. Controlling the surface or interface evolution would be desirable for many of these applications. We attack this problem by considering the bulk contribution, which defines a driving force for the geometric evolution equation, as a distributed control. In order to solve the control problem we use a phase-field approximation and demonstrate the applicability of the approach on various examples. In the first example the effect of an electric field on the evolution of nanostructures on crystalline surfaces is considered. The mathematical problem corresponds to surface diffusion or a Cahn-Hilliard model. In the second example we consider mean curvature flow or a Allen-Cahn model

A Finite Element Framework for Burton-Cabrera-Frank Equation

A finite element framework is presented for the Burton-Cabrera-Frank (BCF) equation. The model is a 2 + 1-dimensional step flow model, discrete in the height but continuous in the lateral directions. The problem consists of adatom diffusion equations on terraces of different atomic height; boundary conditions at steps (terrace boundaries); and a normal velocity law for the motion of such boundaries determined by a two-sided flux, together with one-dimensional edge-diffusion. Two types of boundary conditions, modeling either diffusion limited growth or growth governed by attachment-detachment kinetics at the steps, are considered. We review the basic ideas of the algorithms, already described in [1, 2] and extent it to incorporate anisotropy of the step free energy, the edge mobility and the kinetic coefficients (attachment-detachment rates). The problem is solved using two independent meshes: a two-dimensional mesh for the adatom diffusion and a one-dimensional mesh for the step dynamics governed by an anisotropic geometric evolution law. Finally results on the anisotropic growth of single layer islands are presented.

Simulation of Ostwald Ripening in Homoepitaxy

Ostwald ripening in homoepitaxy in the submonolayer regime is studied by means of numerical simulations. The simulations indicate, that the coarsening kinetics of the average island radius is described by a t1/a power law, where 2 ≤ a ≤ 3. Here a approaches 2, if the ripening is purely kinetics limited (low attachment rate at the island boundaries) and increases with increasing attachment rate — taking the value a = 3 if the ripening is purely diffusion limited (infinite attachment rate at the island boundaries). For the two limiting cases the classical LSW theory is reviewed and compared with the numerical simulations. Besides the scaling law we also investigate the asymptotic scaled island size distribution function and analyse the influence of anisotropic edge energies and the effect of edge diffusion.