Andreas Rätz

Surface evolution of elastically stressed films under deposition by a diffuse interface model

We consider the heteroepitaxial growth of thin films by numerical simulations within a diffuse interface model. The model is applicable to describe the self-organization of nanostructures. The influence of strain, surface energies and kinetics on the surface evolution is considered. A matched asymptotic analysis shows the formal convergence of an anisotropic viscous Cahn-Hilliard model to a general surface evolution equation. The system is solved by adaptive finite elements in three dimensions and in special cases compared with sharp interface models.

Nucleation and growth by a phase field crystal (PFC) model

We review the derivation of a phase field crystal (PFC) model from classical density functional theory (DFT). Through a gradient flow of the Helmholtz free energy functional and appropriate approximations of the correlation functions, higher order nonlinear equations are derived for the evolution of a time averaged density. The equation is solved by finite elements using a semi-implicit time discretization.

Turing instabilities in a mathematical model for signaling networks

GTPase molecules are important regulators in cells that continuously run through an activation/deactivation and membrane-attachment/membrane-detachment cycle. Activated GTPase is able to localize in parts of the membranes and to induce cell polarity. As feedback loops contribute to the GTPase cycle and as the coupling between membrane-bound and cytoplasmic processes introduces different diffusion coefficients a Turing mechanism is a natural candidate for this symmetry breaking. We formulate a mathematical model that couples a reaction–diffusion system in the inner volume to a reaction–diffusion system on the membrane via a flux condition and an attachment/detachment law at the membrane. We present a reduction to a simpler non-local reaction–diffusion model and perform a stability analysis and numerical simulations for this reduction. Our model in principle does support Turing instabilities but only if the lateral diffusion of inactivated GTPase is much faster than the diffusion of activated GTPase.

A diffuse-interface approximation for step flow in epitaxial growth

We consider a step-flow model for epitaxial growth, as proposed by Burton et al. This type of model is discrete in the growth direction but continuous in the lateral directions. The effect of the Ehrlich–Schwoebel barrier, which limits the attachment rate of adatoms to a step from an upper terrace, is included. Mathematically, this model is a 2+1-dimensional dynamic free boundary problem for the steps.In this paper, we propose a diffuse-interface approximation which reproduces an arbitrary Ehrlich–Schwoebel barrier. This is achieved by introducing a degenerate mobility into the so-called viscous Cahn–Hilliard equation. We relate this modified Cahn–Hilliard equation to the sharp interface model via formal matched asymptotic expansion.

Extracting Grain Boundaries and Macroscopic Deformations from Images on Atomic Scale

Abstract Nowadays image acquisition in materials science allows the resolution of grains at atomic scale. Grains are material regions with different lattice orientation which are frequently in addition elastically stressed. At the same time, new microscopic simulation tools allow to study the dynamics of such grain structures. Single atoms are resolved experimentally as well as in simulation results on the data microscale, whereas lattice orientation and elastic deformation describe corresponding physical structures mesoscopically. A qualitative study of experimental images and simulation results and the comparison of simulation and experiment requires the robust and reliable extraction of mesoscopic properties from the microscopic image data. Based on a Mumford–Shah type functional, grain boundaries are described as free discontinuity sets at which the orientation parameter for the lattice jumps. The lattice structure itself is encoded in a suitable integrand depending on a local lattice orientation and one global elastic displacement. For each grain a lattice orientation and an elastic displacement function are considered as unknowns implicitly described by the image microstructure. In addition the approach incorporates solid–liquid interfaces. The resulting Mumford–Shah functional is approximated with a level set active contour model following the approach by Chan and Vese. The implementation is based on a finite element discretization in space and a step size controlled, regularized gradient descent algorithm.

Symmetry breaking in a bulk–surface reaction–diffusion model for signalling networks

Signaling molecules play an important role for many cellular functions. We investigate here a general system of two membrane reaction–diffusion equations coupled to a diffusion equation inside the cell by a Robin-type boundary condition and a flux term in the membrane equations. A specific model of this form was recently proposed by the authors for the GTPase cycle in cells. We investigate here a putative role of diffusive instabilities in cell polarization. By a linearized stability analysis we identify two different mechanisms. The first resembles a classical Turing instability for the membrane subsystem and requires (unrealistically) large differences in the lateral diffusion of activator and substrate. The second possibility on the other hand is induced by the difference in cytosolic and lateral diffusion and appears much more realistic. We complement our theoretical analysis by numerical simulations that confirm the new stability mechanism and allow to investigate the evolution beyond the regime where the linearization applies.Signalling molecules play an important role for many cellular functions. We investigate here a general system of two membrane reaction–diffusion equations coupled to a diffusion equation inside the cell by a Robin-type boundary condition and a flux term in the membrane equations. A specific model of this form was recently proposed by the authors for the GTPase cycle in cells. We investigate here a putative role of diffusive instabilities in cell polarization. By a linearized stability analysis, we identify two different mechanisms. The first resembles a classical Turing instability for the membrane subsystem and requires (unrealistically) large differences in the lateral diffusion of activator and substrate. On the other hand, the second possibility is induced by the difference in cytosolic and lateral diffusion and appears much more realistic. We complement our theoretical analysis by numerical simulations that confirm the new stability mechanism and allow us to investigate the evolution beyond the regime where the linearization applies.

Two-phase flow equations with a dynamic capillary pressure

We investigate the motion of two immiscible fluids in a porous medium described by a two-phase flow system. In the capillary pressure relation, we include static and dynamic hysteresis. The model is well established in the context of the Richards equation, which is obtained by assuming a constant pressure for one of the two phases. We derive an existence result for this hysteresis two-phase model for non-degenerate permeability and capillary pressure curves. A discretization scheme is introduced and numerical results for fingering experiments are obtained. The main analytical tool is a compactness result for two variables that are coupled by a hysteresis relation.

Spiral growth and step edge barriers

The growth of spiral mounds containing a screw dislocation is compared to the growth of wedding cakes by two-dimensional nucleation. Using phase field simulations and homoepitaxial growth experiments on the Pt(111) surface we show that both structures attain the same large scale shape when a significant step-edge barrier suppresses interlayer transport. The higher vertical growth rate of the spiral mounds on Pt(111) reflects the different incorporation mechanisms for atoms in the top region and can be formally represented by an enhanced apparent step-edge barrier.

A diffuse-interface approximation for surface diffusion including adatoms

We introduce a diffuse-interface approximation for solving partial differential equations on evolving surfaces. The model of interest is a fourth-order geometric evolution equation for a growing surface with an additional diffusive adatom density on the surface. Such models arise in the description of epitaxial growth, where the surface of interest is the solid–vapour interface. The model allows us to handle complex geometries in an implicit manner, by considering an evolution equation for a phase-field variable describing the surface and an evolution equation for an extended adatom concentration on a time-independent domain. Matched asymptotic analysis shows the formal convergence towards the sharp interface model and numerical results based on adaptive finite elements demonstrate the applicability of the approach.

Phase-field model for island dynamics in epitaxial growth

A phase-field model is proposed to describe epitaxial growth. In this model the motion of island boundaries of discrete atomic layers is determined by the time evolution of an introduced phase-field variable. We use formally matched asymptotic expansion to determine the asymptotic limit of vanishing interfacial thickness and show the reduction to classical sharp interface models of Burton–Cabrera–Frank type.