Coating of modern surfaces

Coating processes play an important role in nature and many areas of technological applications. For thin liquid surface tension driven films applications range from the spreading of paint, the motion of the human tear film to the interface dynamics of nanoscale films of polymer or complex liquids like the photoresists or photoactive polymer blends that are spin-coated onto a silicon wafer in the process of the manufacturing of electronic chips or organic solar cells. Apart from capillary forces and viscous dissipation, such liquids often show non-Newtonian behaviour and in addition involve effects such as evaporation or diffusion of surfactants, while on the micro- and nanoscale also the impact of intermolecular forces and slippage will eventually play an important role in the dynamics and morphology of interfaces.

Multifunctional Nano-Structures

For thin solid films the development of nanostructures during epitaxial growth has great potential for the design of novel, multifunctional electronic device structures. One focus of invesigations within the research group 7 is the control of the morphology of growing super lattices of so-called 'Quantum Dots', that offer the possiblity to create composite materials having prescribed electronic as well as optoelectronic properties.

Nano-Structures that are created during dewetting processes lead to functionalisation of surfaces. The control of these processes is therefore of great importance for the production of electro chips or tandem structures of thin film solar cells. Another focus of our investigations is devoted to the mathematical modelling and Analysis of such dewetting processes.



  • H.-Chr. Kaiser, D. Knees, A. Mielke, J. Rehberg, E. Rocca, M. Thomas, E. Valdinoci, eds., PDE 2015: Theory and Applications of Partial Differential Equations, 10 of Discrete and Continuous Dynamical Systems -- Series S, American Institute of Mathematical Science, Springfield, 2017, iv+933 pages, (Collection Published).
    HAGs von Christoph bestätigen lassen

  • B. Wagner, B. Rech, A. Münch, V. Mehrmann, eds., Proceedings of the Workshop Mathematics in Industry: Technologies of Thin Film Solar Cells, WIAS, Berlin, 2010, 68 pages, (Collection Published).

  Articles in Refereed Journals

  • S. Bommer, R. Seemann, S. Jachalski, D. Peschka, B. Wagner, Impact of energy dissipation on interface shapes and on rates for dewetting from liquid substrates, Scientific Reports, 8 (2018), pp. 13295/1--13295/11, DOI 10.1038/s41598-018-31418-1 .
    The dependence of the dissipation on the local details of the flow field of a liquid polymer film dewetting from a liquid polymer substrate is shown, solving the free boundary problem for a two-layer liquid system. As a key result we show that the dewetting rates of such a liquid bi-layer system can not be described by a single power law but shows transient behaviour of the rates, changing from increasing to decreasing behaviour. The theoretical predictions on the evolution of morphology and rates of the free surfaces and free interfaces are compared to measurements of the evolution of the polystyrene(PS)-air, the polymethyl methacrylate (PMMA)-air and the PS-PMMA interfaces using in situ atomic force microscopy (AFM), and they show excellent agreement.

  • S. Bergmann, D.A. Barragan-Yani, E. Flegel, K. Albe, B. Wagner, Anisotropic solid-liquid interface kinetics in silicon: An atomistically informed phase-field model, Modelling and Simulation in Materials Science and Engineering, 25 (2017), pp. 065015/1--065015/20, DOI 10.1088/1361-651X/aa7862 .
    We present an atomistically informed parametrization of a phase-field model for describing the anisotropic mobility of liquid-solid interfaces in silicon. The model is derived from a consistent set of atomistic data and thus allows to directly link molecular dynamics and phase field simulations. Expressions for the free energy density, the interfacial energy and the temperature and orientation dependent interface mobility are systematically fitted to data from molecular dynamics simulations based on the Stillinger-Weber interatomic potential. The temperature-dependent interface velocity follows a Vogel-Fulcher type behavior and allows to properly account for the dynamics in the undercooled melt.

  • M. Dziwnik, A. Münch, B. Wagner, An anisotropic phase-field model for solid-state dewetting and its sharp-interface limit, Nonlinearity, 30 (2017), pp. 1465--1496.
    We propose a phase field model for solid state dewetting in form of a Cahn-Hilliard equation with weakly anisotropic surface energy and a degenerate mobility together with a free boundary condition at the film-substrate contact line. We derive the corresponding sharp interface limit via matched asymptotic analysis involving multiple inner layers. The resulting sharp interface model is consistent with the pure surface diffusion model. In addition, we show that the natural boundary conditions, as indicated from the first variation of the total free energy, imply a contact angle condition for the dewetting front, which, in the isotropic case, is consistent with the well-known Young's equation.

  • R. Huth, S. Jachalski, G. Kitavtsev, D. Peschka, Gradient flow perspective on thin-film bilayer flows, Journal of Engineering Mathematics, 94 (2015), pp. 43--61.
    We study gradient flow formulations of thin-film bilayer flows with triple-junctions between liquid/liquid/air. First we highlight the gradient structure in the Stokes free-boundary flow and identify its solutions with the well known PDE with boundary conditions. Next we propose a similar gradient formulation for the corresponding thin-film model and formally identify solutions with those of the corresponding free-boundary problem. A robust numerical algorithm for the thin-film gradient flow structure is then provided. Using this algorithm we compare the sharp triple-junction model with precursor models. For their stationary solutions a rigorous connection is established using Gamma-convergence. For time-dependent solutions the comparison of numerical solutions shows a good agreement for small and moderate times. Finally we study spreading in the zero-contact angle case, where we compare numerical solutions with asymptotically exact source-type solutions.

  • M.D. Korzec, P. Evans, From bell shapes to pyramids: A reduced continuum model for self-assembled quantum dot growth, Physica D. Nonlinear Phenomena, 239 (2010), pp. 465--474.

  • D. Peschka, A. Münch, B. Niethammer, Thin film rupture for large slip, Journal of Engineering Mathematics, 66 (2010), pp. 33--51.
    This paper studies the rupture of thin liquid films on hydrophobic substrates, assuming large slip at the liquidsolid interface. Using a recently developed em strong slip lubrication model, it is shown that the rupture passes through up to three self-similar regimes with different dominant balances and different scaling exponents. For one of these regimes the similarity is of second kind, and the similarity exponent is determined by solving a boundary value problem for a nonlinear ODE. For this regime we also prove finite-time rupture.

  • D. Peschka, A. Münch, B. Niethammer, Self-similar rupture of viscous thin films in the strong-slip regime, Nonlinearity, 23 (2010), pp. 409--427.
    We consider rupture of thin viscous films in the strong-slip regime with small Reynolds numbers. Numerical simulations indicate that near the rupture point viscosity and van-der-Waals forces are dominant and that there are self-similar solutions of the second kind. For a corresponding simplified model we rigorously analyse self-similar behaviour. There exists a one-parameter family of self-similar solutions and we establish necessary and sufficient conditions for convergence to any self-similar solution in a certain parameter regime. We also present a conjecture on the domains of attraction of all self-similar solutions which is supported by numerical simulations.

  • M.D. Korzec, P.L. Evans, A. Münch, B. Wagner, Stationary solutions of driven fourth- and sixth-order Cahn--Hilliard type equations, SIAM Journal on Applied Mathematics, 69 (2008), pp. 348-374.
    New types of stationary solutions of a one-dimensional driven sixth-order Cahn-Hilliard type equation that arises as a model for epitaxially growing nano-structures such as quantum dots, are derived by an extension of the method of matched asymptotic expansions that retains exponentially small terms. This method yields analytical expressions for far-field behavior as well as the widths of the humps of these spatially non-monotone solutions in the limit of small driving force strength which is the deposition rate in case of epitaxial growth. These solutions extend the family of the monotone kink and antikink solutions. The hump spacing is related to solutions of the Lambert $W$ function. Using phase space analysis for the corresponding fifth-order dynamical system, we use a numerical technique that enables the efficient and accurate tracking of the solution branches, where the asymptotic solutions are used as initial input. Additionally, our approach is first demonstrated for the related but simpler driven fourth-order Cahn-Hilliard equation, also known as the convective Cahn-Hilliard equation.

  • M. Rauscher, R. Blossey, A. Münch, B. Wagner, Spinodal dewetting of thin films with large interfacial slip: Implications from the dispersion relation, Langmuir, 24 (2008), pp. 12290-12294.

  • R. Fetzer, A. Münch, B. Wagner, M. Rauscher, K. Jacobs, Quantifying hydrodynamic slip: A comprehensive analysis of dewetting profiles, Langmuir, 23 (2007), pp. 10559-10566.
    To characterize non-trivial boundary conditions of a liquid flowing past a solid, the slip length is commonly used as a measure. From the profile of a retracting liquid front as measured, e.g., with atomic force microscopy, the slip length as well as the capillary number can be extracted by the help of the Stokes model for a thin liquid film dewetting from a solid substrate. Specifically, we use a lubrication model derived from the Stokes model for strong slippage and linearize the film profile around the flat, unperturbed film, and, for small slip lengths a Taylor approximation of the linearisation for the full Stokes model. Furthermore, from the capillary number and the knowledge of the liquid front velocity and the surface tension, we can obtain the viscosity of the fluid film. We compare theoretical and experimental results, test the consistency and the validity of the models/approximations, and give an easy-to-follow manual of how they can be used to analyze experiments.

  • R. Fetzer, M. Rauscher, A. Münch, B. Wagner, K. Jacobs, Slip-controlled thin film dynamics, Europhysics Letters, 75 (2006), pp. 638-644.
    In this study, we present a novel method to assess the slip length and the viscosity of thin films of highly viscous Newtonian liquids. We quantitatively analyse dewetting fronts of low molecular weight polystyrene melts on Octadecyl- (OTS) and Dodecyltrichlorosilane (DTS) polymer brushes. Using a thin film (lubrication) model derived in the limit of large slip lengths, we can extract slip length and viscosity. We study polymer films with thicknesses between 50 nm and 230 nm and various temperatures above the glass transition. We find slip lengths from 100 nm up to 1 $mu$m on OTS and between 300 nm and 10 $mu$m on DTS covered silicon wafers. The slip length decreases with temperature. The obtained values for the viscosity are consistent with independent measurements.

  • P. Evans, A. Münch, Interaction of advancing fronts and meniscus profiles formed by surface-tension-gradient-driven liquid films, SIAM Journal on Applied Mathematics, 66 (2006), pp. 1610-1631.

  • R. Fetzer, K. Jacobs, A. Münch, B. Wagner, T.P. Witelski, New slip regimes and the shape of dewetting thin liquid films, Physical Review Letters, 95 (2005), pp. 127801/1-127801/4.

  • A. Münch, Dewetting rates of thin liquid films, Physics of Fluids, 17 (2005), pp. S309--S318.

  • M. Rauscher, A. Münch, B. Wagner, A thin-film equation for viscoelastic liquids of Jeffreys type, The European Physical Journal. E. Soft Matter, 17 (2005), pp. 373--379.

  Preprints, Reports, Technical Reports

  • D. Peschka, M. Thomas, T. Ahnert, A. Münch, B. Wagner, Gradient structures for flows of concentrated suspensions, Preprint no. 2543, WIAS, Berlin, 2018, DOI 10.20347/WIAS.PREPRINT.2543 .
    Abstract, PDF (6456 kByte)
    In this work we investigate a two-phase model for concentrated suspensions. We construct a PDE formulation using a gradient flow structure featuring dissipative coupling between fluid and solid phase as well as different driving forces. Our construction is based on the concept of flow maps that also allows it to account for flows in moving domains with free boundaries. The major difference compared to similar existing approaches is the incorporation of a non-smooth two-homogeneous term to the dissipation potential, which creates a normal pressure even for pure shear flows

  • D. Peschka, S. Haefner, K. Jacobs, A. Münch, B. Wagner, Signatures of slip in dewetting polymer films, Preprint no. 2538, WIAS, Berlin, 2018, DOI 10.20347/WIAS.PREPRINT.2538 .
    Abstract, PDF (7811 kByte)
    Thin liquid polymer films on hydrophobic substrates are susceptable to rupture and formation of holes, which in turn initiate a complex dewetting process that eventually evolves into characteristic stationary droplet patterns. Experimental and theoretical studies suggest that the specific type of droplet pattern largely depends on the nature of the polymer-substrate boundary condition. To follow the morphological evolution numerically over long time scales and for the multiple length scales involved has so far been a major challenge. In this study a highly adaptive finite-element based numerical scheme is presented that allows for large-scale simulations to follow the evolution of the dewetting process deep into the nonlinear regime of the model equations, capturing the complex dynamics including shedding of droplets. In addition, the numerical results predict the previouly unknown shedding of satellite droplets during the destabilisation of liquid ridges, that form during the late stages of the dewetting process. While the formation of satellite droplets is well-known in the context of elongating fluid filaments and jets, we show here that for dewetting liquid ridges this property can be dramatically altered by the interfacial condition between polymer and substrate, namely slip.

  Talks, Poster

  • M. Thomas, D. Peschka, B. Wagner, V. Mehrmann, M. Rosenau, Modeling and analysis of suspension flows, MATH+ Center Days 2018, October 31 - November 2, 2018, Zuse-Institut Berlin (ZIB), Berlin, October 31, 2018.

  • D. Peschka, Motion of thin droplets over surfaces, Making a Splash -- Driplets, Jets and Other Singularities, March 20 - 24, 2017, Brown University, Institute for Computational and Experimental Research in Mathematics (ICERM), Providence, USA, March 22, 2017.

  • D. Peschka, Variational structure of fluid motion with contact lines in thin-film models, Kolloquium Angewandte Mathematik, Universität der Bundeswehr, München, May 31, 2017.

  • D. Peschka, Multi-phase flows with contact lines: Solid vs liquid substrates, Industrial and Applied Mathematics Seminar, University of Oxford, Mathematical Institute, UK, October 27, 2016.

  • D. Peschka, Thin film free boundary problems --- Modeling of contact line dynamics with gradient formulations, CeNoS-Kolloquium, Westfälische Wilhelms-Universität Münster, Center for Nonlinear Science, January 12, 2016.

  • D. Peschka, Modeling and applications of bilayer flows, Seminar of the Research Training Group GRK 1276 ``Structure Formation and Transport in Complex Systems'', Universität des Saarlandes, Institut für Theoretische Physik, Saarbrücken, January 27, 2015.

  • G. Kitavtsev, L. Recke, B. Wagner, Derivation, analysis and numerics of reduced ODE models describing coarsening dynamics, 3textsuperscriptrd European Postgraduate Fluid Dynamics Conference, Nottingham, UK, July 13 - 16, 2009.

  • G. Kitavtsev, Derivation, analysis and numerics of reduced ODE models describing coarsening dynamics, 3$^rm rd$ European Postgraduate Fluid Dynamics Conference, July 13 - 16, 2009, University of Nottingham, UK, July 15, 2009.

  • G. Kitavtsev, Reduced ODE models describing coarsening dynamics of slipping droplets and a geometrical approach for their derivation, Oberseminar, Universität Bonn, Institut für Angewandte Mathematik, July 23, 2009.

  • D. Peschka, Self-similar rupture for thin films with slip, EUROMECH Colloquium 497 --- Recent Developments and New Directions in Thin-Film Flow, July 6 - 9, 2009, Royal Society of Edinburgh, UK, July 8, 2009.