Rainer Backofen

Derivation of the phase-field-crystal model for colloidal solidification.

The phase-field-crystal model is by now widely used in order to predict crystal nucleation and growth. For colloidal solidification with completely overdamped individual particle motion, we show that the phase-field-crystal dynamics can be derived from the microscopic Smoluchowski equation via dynamical density-functional theory. The different underlying approximations are discussed. In particular, a variant of the phase-field-crystal model is proposed which involves less approximations than the standard phase-field-crystal model. We finally test the validity of these phase-field-crystal models against dynamical density-functional theory. In particular, the velocities of a linear crystal front from the undercooled melt are compared as a function of the undercooling for a two-dimensional colloidal suspension of parallel dipoles. Good agreement is only obtained by a drastic scaling of the free energies in the phase-field-crystal model in order to match the bulk freezing transition point.

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.

A phase-field-crystal approach to critical nuclei

We investigate a phase-field-crystal model for homogeneous nucleation. Instead of obtaining the time evolution of a density field towards equilibrium, we use a string method to identify saddle points in phase space. The saddle points allow us to obtain the nucleation barrier and the critical nucleus. The advantage of using the phase-field-crystal model for this task is that it can be used to resolve atomistic effects. The results obtained indicate different properties of the critical nucleus as compared with those for bulk crystals and provide a detailed description of the nucleation process.

Capturing the complex physics behind universal grain size distributions in thin metallic films

Grain growth experiments on thin metallic films have shown the geometric and topological characteristics of the grain structure to be universal and independent of many experimental conditions. The universal size distribution, however, is found to differ both qualitatively and quantitatively from the standard Mullins curvature driven model of grain growth; with the experiments exhibiting an excess of small grains (termed an ”ear”) and an excess of very large grains (termed a ”tail”) compared with the model. While a plethora of extensions of the Mullins model have been proposed to explain these characteristics, none have been successful. In this work, large scale simulations of a model that resolves the atomic scale on diffusive time scales, the phase field crystal model, is used to examine the complex phenomena of grain growth. The results are in remarkable agreement with the experimental results, recovering the characteristic ”ear” and ”tail” features of the experimental grain size distribution. The simulations also indicate that while the geometric and topological characteristics are universal, the dynamic growth exponent is not.

The influence of membrane bound proteins on phase separation and coarsening in cell membranes

A theoretical explanation of the existence of lipid rafts in cell membranes remains a topic of lively debate. Large, micrometer sized rafts are readily observed in artificial membranes and can be explained using thermodynamic models for phase separation and coarsening. In live cells such domains are not observed and various models are proposed to describe why the systems do not coarsen. We review these attempts critically and show within a phase field approach that membrane bound proteins have the potential to explain the different behaviour observed in vitro and in vivo. Large scale simulations are performed to compute scaling laws and size distribution functions under the influence of membrane bound proteins and to observe a significant slow down of the domain coarsening at longer times and a breakdown of the self-similarity of the size-distribution function.

Particles on curved surfaces: a dynamic approach by a phase-field-crystal model.

We present a dynamic model to study ordering of particles on arbitrary curved surfaces. Thereby the particles are represented as maxima in a density field and a surface partial differential equation for the density field is solved to the minimal energy configuration. We study annihilation of dislocations within the ordered system and premelting along grain-boundary scars. The obtained minimal energy configurations on a sphere are compared with existing results and scaling laws are computed for the number of excess dislocations as a function of system size.

A Continuous Approach to Discrete Ordering on S^2

We consider the classical problem to find optimal distributions of interacting particles on a sphere by solving an evolution problem for a particle density. The higher order surface partial differential equation is an approximation of a surface dynamic density functional theory. We motivate the approach phenomenologically and sketch a derivation of the model starting from an interatomic potential. Different numerical approaches are discussed to solve the evolution problem: (a) an implicit approach to describe the surface using a phase-field description, (b) a parametric finite element approach, and (c) a spectral method based on nonequispaced fast Fourier transforms on the sphere. Results for computed minimal energy configurations are discussed for various particle numbers and are compared with known rigorous asymptotic results. Furthermore extensions to other more complex and evolving surfaces are mentioned.

Engineered Coalescence by Annealing 3D Ge Microstructures into High-Quality Suspended Layers on Si.

The move from dimensional to functional scaling in microelectronics has led to renewed interest toward integration of Ge on Si. In this work, simulation-driven experiments leading to high-quality suspended Ge films on Si pillars are reported. Starting from an array of micrometric Ge crystals, the film is obtained by exploiting their temperature-driven coalescence across nanometric gaps. The merging process is simulated by means of a suitable surface-diffusion model within a phase-field approach. The successful comparison between experimental and simulated data demonstrates that the morphological evolution is driven purely by the lowering of surface-curvature gradients. This allows for fine control over the final morphology to be attained. At fixed annealing time and temperature, perfectly merged films are obtained from Ge crystals grown at low temperature (450 °C), whereas some void regions still persist for crystals grown at higher temperature (500 °C) due to their different initial morphology. The latter condition, however, looks very promising for possible applications. Indeed, scanning tunneling electron microscopy and high-resolution transmission electron microscopy analyses show that, at least during the first stages of merging, the developing film is free from threading dislocations. The present findings, thus, introduce a promising path to integrate Ge layers on Si with a low dislocation density.

Complex dewetting scenarios of ultrathin silicon films for large-scale nanoarchitectures

Si-based nanoarchitectures are formed with unprecedented precision and reproducibility via templated dewetting of thin SOI. Dewetting is a ubiquitous phenomenon in nature; many different thin films of organic and inorganic substances (such as liquids, polymers, metals, and semiconductors) share this shape instability driven by surface tension and mass transport. Via templated solid-state dewetting, we frame complex nanoarchitectures of monocrystalline silicon on insulator with unprecedented precision and reproducibility over large scales. Phase-field simulations reveal the dominant role of surface diffusion as a driving force for dewetting and provide a predictive tool to further engineer this hybrid top-down/bottom-up self-assembly method. Our results demonstrate that patches of thin monocrystalline films of metals and semiconductors share the same dewetting dynamics. We also prove the potential of our method by fabricating nanotransfer molding of metal oxide xerogels on silicon and glass substrates. This method allows the novel possibility of transferring these Si-based patterns on different materials, which do not usually undergo dewetting, offering great potential also for microfluidic or sensing applications.

Stress Induced Branching of Growing Crystals on Curved Surfaces.

If two-dimensional crystals grow on a curved surface, the Gaussian curvature of the surface induces elastic stress and affects the growth pathway. The elastic stress can be alleviated by incorporating defects or, if this is energetically unfavorable, via an elastic instability which leads to anisotropic growth with branched ribbonlike structures. This instability provides a generic route to grow defect-free crystals on curved surfaces. Depending on the elastic properties of the crystal and the geometric properties of the surface, different growth morphologies with two-, four-, and sixfold symmetry develop. Using a phase field crystal type modeling approach, we provide a microscopic understanding of the morphology selection.