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 spincoated 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 nonNewtonian 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 NanoStructures
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 socalled 'Quantum Dots', that offer the possiblity to create composite materials having prescribed electronic as well as optoelectronic properties.
NanoStructures 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.
Publications
Monographs

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).

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

D. Peschka, S. Haefner, L. Marquant, K. Jacobs, A. Münch, B. Wagner, Signatures of slip in dewetting polymer films, Proceedings of the National Academy of Sciences of the United States of America, 116 (2019), pp. 92759284, DOI 10.1073/pnas.1820487116 .

D. Peschka, Variational approach to dynamic contact angles for thin films, Physics of Fluids, 30 (2018), pp. 082115/1082115/11, DOI 10.1063/1.5040985 .
Abstract
This paper investigates a variational approach to viscous flows with contact line dynamics based on energydissipation modeling. The corresponding model is reduced to a thinfilm equation and its variational structure is also constructed and discussed. Feasibility of this modeling approach is shown by constructing a numerical scheme in 1D and by computing numerical solutions for the problem of gravity driven droplets. Some implications of the contact line model are highlighted in this setting. 
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. 13295/113295/11, DOI 10.1038/s41598018314181 .
Abstract
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 twolayer liquid system. As a key result we show that the dewetting rates of such a liquid bilayer 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 PSPMMA interfaces using in situ atomic force microscopy (AFM), and they show excellent agreement. 
S. Bergmann, D.A. BarraganYani, E. Flegel, K. Albe, B. Wagner, Anisotropic solidliquid interface kinetics in silicon: An atomistically informed phasefield model, Modelling and Simulation in Materials Science and Engineering, 25 (2017), pp. 065015/1065015/20, DOI 10.1088/1361651X/aa7862 .
Abstract
We present an atomistically informed parametrization of a phasefield model for describing the anisotropic mobility of liquidsolid 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 StillingerWeber interatomic potential. The temperaturedependent interface velocity follows a VogelFulcher type behavior and allows to properly account for the dynamics in the undercooled melt. 
M. Dziwnik, A. Münch, B. Wagner, An anisotropic phasefield model for solidstate dewetting and its sharpinterface limit, Nonlinearity, 30 (2017), pp. 14651496.
Abstract
We propose a phase field model for solid state dewetting in form of a CahnHilliard equation with weakly anisotropic surface energy and a degenerate mobility together with a free boundary condition at the filmsubstrate 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 wellknown Young's equation. 
R. Huth, S. Jachalski, G. Kitavtsev, D. Peschka, Gradient flow perspective on thinfilm bilayer flows, Journal of Engineering Mathematics, 94 (2015), pp. 4361.
Abstract
We study gradient flow formulations of thinfilm bilayer flows with triplejunctions between liquid/liquid/air. First we highlight the gradient structure in the Stokes freeboundary flow and identify its solutions with the well known PDE with boundary conditions. Next we propose a similar gradient formulation for the corresponding thinfilm model and formally identify solutions with those of the corresponding freeboundary problem. A robust numerical algorithm for the thinfilm gradient flow structure is then provided. Using this algorithm we compare the sharp triplejunction model with precursor models. For their stationary solutions a rigorous connection is established using Gammaconvergence. For timedependent solutions the comparison of numerical solutions shows a good agreement for small and moderate times. Finally we study spreading in the zerocontact angle case, where we compare numerical solutions with asymptotically exact sourcetype solutions. 
M.D. Korzec, P. Evans, From bell shapes to pyramids: A reduced continuum model for selfassembled quantum dot growth, Physica D. Nonlinear Phenomena, 239 (2010), pp. 465474.

D. Peschka, A. Münch, B. Niethammer, Thin film rupture for large slip, Journal of Engineering Mathematics, 66 (2010), pp. 3351.
Abstract
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 selfsimilar 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 finitetime rupture. 
D. Peschka, A. Münch, B. Niethammer, Selfsimilar rupture of viscous thin films in the strongslip regime, Nonlinearity, 23 (2010), pp. 409427.
Abstract
We consider rupture of thin viscous films in the strongslip regime with small Reynolds numbers. Numerical simulations indicate that near the rupture point viscosity and vanderWaals forces are dominant and that there are selfsimilar solutions of the second kind. For a corresponding simplified model we rigorously analyse selfsimilar behaviour. There exists a oneparameter family of selfsimilar solutions and we establish necessary and sufficient conditions for convergence to any selfsimilar solution in a certain parameter regime. We also present a conjecture on the domains of attraction of all selfsimilar solutions which is supported by numerical simulations. 
M.D. Korzec, P.L. Evans, A. Münch, B. Wagner, Stationary solutions of driven fourth and sixthorder CahnHilliard type equations, SIAM Journal on Applied Mathematics, 69 (2008), pp. 348374.
Abstract
New types of stationary solutions of a onedimensional driven sixthorder CahnHilliard type equation that arises as a model for epitaxially growing nanostructures 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 farfield behavior as well as the widths of the humps of these spatially nonmonotone 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 fifthorder 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 fourthorder CahnHilliard equation, also known as the convective CahnHilliard 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. 1229012294.

R. Fetzer, A. Münch, B. Wagner, M. Rauscher, K. Jacobs, Quantifying hydrodynamic slip: A comprehensive analysis of dewetting profiles, Langmuir, 23 (2007), pp. 1055910566.
Abstract
To characterize nontrivial 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 easytofollow manual of how they can be used to analyze experiments. 
R. Fetzer, M. Rauscher, A. Münch, B. Wagner, K. Jacobs, Slipcontrolled thin film dynamics, Europhysics Letters, 75 (2006), pp. 638644.
Abstract
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 surfacetensiongradientdriven liquid films, SIAM Journal on Applied Mathematics, 66 (2006), pp. 16101631.

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. 127801/1127801/4.

A. Münch, Dewetting rates of thin liquid films, Physics of Fluids, 17 (2005), pp. S309S318.

M. Rauscher, A. Münch, B. Wagner, A thinfilm equation for viscoelastic liquids of Jeffreys type, The European Physical Journal. E. Soft Matter, 17 (2005), pp. 373379.
Contributions to Collected Editions

D. Peschka, M. Thomas, T. Ahnert, A. Münch, B. Wagner, Gradient structures for flows of concentrated suspensions, in: Topics in Applied Analysis and Optimisation, M. Hintermüller, J.F. Rodrigues, eds., CIM Series in Mathematical Sciences, Springer Nature Switzerland AG, Cham, 2019, pp. 295318, DOI 10.1007/9783030331160 .
Abstract
In this work we investigate a twophase 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 nonsmooth twohomogeneous term to the dissipation potential, which creates a normal pressure even for pure shear flows.
Preprints, Reports, Technical Reports

A. Mielke, R.R. Netz, S. Zendehroud, A rigorous derivation and energetics of a wave equation with fractional damping, Preprint no. 2718, WIAS, Berlin, 2020, DOI 10.20347/WIAS.PREPRINT.2718 .
Abstract, PDF (312 kByte)
We consider a linear system that consists of a linear wave equation on a horizontal hypersurface and a parabolic equation in the half space below. The model describes longitudinal elastic waves in organic monolayers at the waterair interface, which is an experimental setup that is relevant for understanding wave propagation in biological membranes. We study the scaling regime where the relevant horizontal length scale is much larger than the vertical length scale and provide a rigorous limit leading to a fractionallydamped wave equation for the membrane. We provide the associated existence results via linear semigroup theory and show convergence of the solutions in the scaling limit. Moreover, based on the energydissipation structure for the full model, we derive a natural energy and a natural dissipation function for the fractionallydamped wave equation with a time derivative of order 3/2.
Talks, Poster

D. Peschka, Mathematical modeling and simulation of flows and the interaction with a substrate using energetic variational methods, Vortrag im Rahmen des SFB1194, Technische Universität Darmstadt, January 22, 2020.

D. Peschka , Variational modeling of bulk and interface effects in fluid dynamics, SPP 2171 Advanced School ``Introduction to Wetting Dynamics'', February 17  21, 2020, Westfälische WilhelmsUniversität Münster, February 18, 2020.

M.H. Farshbaf Shaker, D. Peschka, M. Thomas, Modeling and analysis of suspension flows, Visit of the Scientific Advisory Board of MATH+, November 11, 2019.

M.H. Farshbaf Shaker, D. Peschka, M. Thomas, Modeling and analysis of suspension flows, 1st MATH+ Day, Berlin, December 13, 2019.

D. Peschka, Dynamic contact angles via generalized gradient flows, Modelling of Thin Liquid Films  Asymptotic Approach vs. Gradient Dynamics, April 28  May 3, 2019, Banff International Research Station for Mathematical Information and Discovery, Canada, April 30, 2019.

D. Peschka, Dynamic contact angles via gradient flows, 694. WEHeraeusSeminar ``Wetting on Soft or Microstructured Surfaces'', Bad Honnef, April 10  13, 2019.

D. Peschka, Gradient formulations with flow maps  Mathematical and numerical approaches to free boundary problems, Kolloquium des Graduiertenkollegs 2339 ``Interfaces, Complex Structures, and Singular Limits'', Universität Regensburg, May 24, 2019.

D. Peschka, Mathematical modeling and simulation of substrateflow interaction using generalized gradient flow, Begutachtungskolloquium für die Anträge des SPP 2171 ``Dynamische Benetzung flexibler, adaptiver und schaltbarer Oberflächen'', Mainz, February 7  8, 2019.

D. Peschka, Mathematical modeling of fluid flows using gradient systems, Seminar in PDE and Applications, Delft University of Technology, Netherlands, May 28, 2019.

D. Peschka, Steering pattern formation of viscous flows, DMVJahrestagung 2019, Sektion ``Differentialgleichungen und Anwendungen'', September 23  26, 2019, KIT  Karlsruher Institut für Technologie, September 23, 2019.

A. Münch, B. Wagner, Nonlinear viscoelastic effects of polymer and hydrogel layers sliding on liquid substrates, 694. WEHeraeusSeminar, Bad Honnef, April 11  13, 2019.

M. Thomas, GENERIC structures with bulkinterface interaction, SFB 910 Symposium ``Energy Based Modeling, Simulation and Control'', October 25, 2019, Technische Universität Berlin, October 25, 2019.

M. Thomas, Gradient structures for flows of concentrated suspensions, 9th International Congress on Industrial and Applied Mathematics (ICIAM 2019), Thematic Minisymposium MS ME775 ``Recent Advances in Understanding Suspensions and Granular Media Flow'', July 15  19, 2019, Valencia, Spain, July 17, 2019.

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, ZuseInstitut 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 thinfilm models, Kolloquium Angewandte Mathematik, Universität der Bundeswehr, München, May 31, 2017.

D. Peschka, Multiphase 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, CeNoSKolloquium, Westfälische WilhelmsUniversität 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, Selfsimilar rupture for thin films with slip, EUROMECH Colloquium 497  Recent Developments and New Directions in ThinFilm Flow, July 6  9, 2009, Royal Society of Edinburgh, UK, July 8, 2009.
Contact
Mathematical Context
 Analysis of Partial Differential Equations and Evolutionary Equations
 Free boundary problems for partial differential equations
 Functional analysis and operator theory
 Modeling, analysis and numerics of phase field models
 Multi scale modeling and hybrid models
 Multiscale Modeling and Asymptotic Analysis
 Systems of partial differential equations: modeling, numerical analysis and simulation
 Variational methods