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Project C10 |
Modelling, asymptotic analysis and numerical simulation of the dynamics of thin film nanostructures on crystal surfaces | |
| Project heads: | PD Dr. Andreas Münch, HU, WIAS (until September 2008); now at the Mathematical Insitute, Oxford, U.K. |
| PD Dr. Barbara Wagner , WIAS (2nd funding period). | |
| Support: | Matheon DFG Research Centre |
| Research Team: | Dr. P. L. Evans, HU (since January 2004) and |
| M. Korzec, WIAS (since August 2006) | |
| Publications: | See separate publications list. |
| Contact: | For further information about this project, please contact Dr Wagner. |
Thin films play an important role in many application areas and also in everyday life.
Mathematical models of liquid films have long made use of the small slope of the thin film surface to achieve a model reduction from the underlying 3-D Navier-Stokes free boundary problem to a corresponding lubrication model. Due to the influence of surface tension at the free liquid-air interface, these models consist of nonlinear, degenerate partial differential equations of fourth order, or of systems of such equations. Besides mean surface tension, these models include the effects of a variety of forces, such as gravity or Marangoni shear stresses or intermolecular forces. The interplay of these forces leads to rich phenomena, which include nonlinear wave motion such as new types of nonclassical shock waves, pinch-off and hole formation, or pattern formation such as the fingering at the front of a driven liquid film. [Mue03]
Thin solid films that are epitaxially grown on a solid substrate are also susceptible to morphological instabilities, most notably of Asaro-Tiller-Grinfeld type. During heteroepitaxy, that is, deposition of a single-crystal layer of one material (such as germanium) on top of a single crystal substrate of a different material (e.g. silicon), there is a mismatch between the lattice constants in the thin film and the substrate. This causes elastic stresses and may lead to instabilities driven by surface diffusion, whereby the elastic energy in the film has a destabilizing and the surface energy a stabilizing effect. One possible, and technologically important, result is the formation of Stranski-Krastanov islands, also called quantum dots (QDs), on top of a wetting layer. (See, for example, experiments at IBM on films of Ge on Si(001).) In the case of solid films, the orientation-dependence of the surface energy promotes facetting of the developing film.
A similar approach unifies our study of these two topics. The small slope approximation, shown to be the key necessary for a systematic study of many thin liquid film problems, has only been applied recently to thin epitaxially grown films. It leads to to high-order partial differential equations for the evolving surface. As there is but a single evolution equation, with one space dimension fewer than the original problem, this represents a significant model reduction. Furthermore, in both cases a wetting layer arising from intermolecular forces has a strong influence on film development.
While the development and systematic study of such high-order model equations require new mathematical theory as well as numerical methods, it also offers the possibility of understanding and controlling the dynamical processes leading to the experimentally observed long-time patterns. In the case of epitaxially grown films, this may even allow the design of superlattices of quantum dots, having very different electronic as well as optoelectronic properties.
Epitaxial growth of nanostructures was considered via a small-slope approximation. In our first paper in this area, we identified new stationary solutions of a sixth-order Cahn-Hilliard type PDE describing the orientation of the film surface [KEMW07]. These were found using an extension of the method of matched asymptotic expansions that retains exponentially small terms. Using these we followed the solution branches by a continuation technique.
Recent work revolves around investigation of a new small-slope model, based on an extension of the existing model by Tekalign and Spencer [J. Appl. Phys., 96(10):5505-5512 (2004)] to include an anisotropic surface energy [KE09]. We are working on studies of stability, coarsening rates and, eventually, evolution of arrays of QDs in three dimensions.
Numerical work with similar models has been performed by others e.g. Chiu's group [e.g. C.-H. Chiu and Z. Huang, J. Appl. Phys., 101:113540 (2007)], but these involve large-scale FEM simulations. With the thin film approach we hope to achieve comparable results at lower computational cost.
| Observation of a dewetting polymer film. A rim forms, surrounding a dewetted area about 60 μm in diameter. The rim develops an instability. (Fetzer et al.) | Computed stationary solution for a facetted solid film. |
| Profiles from 3D simulations of quantum dots, with constant surface energy (G=0) and an anisotropic surface energy function (G=0.25). | |
Our group has completed considerable work on non-Newtonian rheology and slip. We studied films of polymer liquids of thicknesses ranging from tens to a few hundred nanometers on hydrophobized silicon wafers. These films tend to recoil from the substrate in a dewetting process that is initiated by the formation of holes. The film thickness is reduced from its initial thickness to a residual film of no more than a few nanometres. At these scales intermolecular forces drive the increase of the dewetted area. The process of dewetting and the eventual patterns of droplets have attracted considerable experimental and theoretical research in the physics (and surface chemistry) community. Nevertheless, many of the experimental observations are not yet well understood.
Further progress is needed on the mathematical side, by systematically investigating and setting up new lubrication models that describe the dewetting dynamics. The impact of the small scale shows that slip plays a crucial role in the dewetting dynamics. We derived a whole family of lubrication models from the Navier-Stokes free boundary problem with a Navier slip condition at the interface for different regimes of the slip length [MWW05].
Investigation of the solution structure resulted in the prediction of a transition from a spatially oscillating profile to a monotonic decaying one for the dewetting rim. Our results agree both qualitatively and quantitatively with experimental results, and have now led to a new method to measure slip lengths based on the morphology of the rim [FJMWW05, FMWRJ07].
Our work on rim propagation and instability of the contact line revealed that slip along the substrate may play a key role for this instability. Our numerical simulations and stability analysis recover the characteristic asymmetric growth of the finger instability if the liquid film is allowed to slip [MW04]. These numerical results have then been supported by a systematic asymptotic investigation, and derivation and analysis of sharp-interface models [EKM06, KMW06, KMW07].
The new lubrication models for nanoscale dynamics are also investigated on a more rigorous basis via two of our doctoral students (Dirk Peschka and Georgy Kitavtsev in the GRK 1128 Research Training Group), considering the early phase of the problem, rupture, and the late phase behaviour of coarsening.
Work on Marangoni-driven films investigated the dynamics of a film rising from a reservoir onto a tilted substrate, in particular, near the meniscus that connects the film to the reservoir. Equilibrium solutions for the meniscus at different inclination angles of the substrate using phase space methods were found and a rich solution structure could be classified, including a monotonic equilibrium solution and multiple non-monotonic solutions with the same far-field film thickness [ME04]. A second paper [EM05] explored the interaction of these solutions with the classical and non-classical wave structures in the thin film.
The experience gained from this work benefitted an adjunct project, where the flow of a thin polymer film on a vertical rotating disk that drags the liquid out of a reservoir was investigated. It resulted in finite-element code for a generalised lubrication equation in two space dimensions [AMW07]. The expertise of our group also resulted in collaboration with an industrial partner and a grant supporting a doctoral student (Ernst Höschele) for three years.
A brief summary of results from the first funding period is also available.
In addition we were involved in the organization of the "Thin Film Solar Cells" Matheon industry workshop, held in Berlin in October 2008. Our work is expected to have potential in new thin film solar cells and printed electronics.
A list of project publications is available.