Thin Film Equations

The mathematical modeling of the dynamics of thin liquid as well as thin solid films typically uses the small ratio of vertical to horizontal scales of the evolving morphologies to simplify the underlying free boundary problems to dimension-reduced partial differential equations for evolving free surface and/or interfaces of the thin films.

Methods of singular perturbation theory show that the resulting partial differential equations are typically of high order, such as lubrication type equations, or systems thereof, or convective Cahn-Hilliard type equations.

To understand the complex nonlinear solution structure of these equations and to complement numerical simulation, matched asymptotic expansions are being derived and exponential asymptotics are extended for these higher order equations. Both, formal and rigorous asymptotic analysis lead to corresponding sharp-interface models that are used to systematically describe important properties, such as the stability of moving contact lines or the long-time behaviour of coarsening binary alloys, the long-time convergence to the self-similar approach towards finite-time blow-up, describing the process of film rupture, or the discovery of novel stationary solutions describing the facetting of so-called quantum dots.

Sharp interface limits of generalized Navier-Stokes-Korteweg systems

Sharp interface limits are considered for isothermal and non-isothermal Korteweg and Navier-Stokes-Korteweg phase field models.

The sharp interface limit is obtained by matched asymptotic expansions of the phase fields in powers of the interface width ε. These expansions are considered in the interfacial region (inner expansion) and in the bulk (outer expansion), and are matched order by order. This results in partial differential equations for the diffuse fields and a series of boundary conditions at the interface.

For different scalings, solvability criteria for the inner equations are established. This leads to various settings of sharp interface models with different jump conditions at the sharp interface.

Sharp limits of regularized diffusion equations with mechanical coupling for applications in energy technology

Two future-oriented energy storage systems, i.e. the storage of electrical energy in lithium-ion batteries and the storage of hydrogen in metal hydrides, are studied. In both applications foreign atoms, lithium respectively hydrogen, will reversibly be stored in crystals. During loading and unloading the foreign atoms form a two phase system with high and low concentration. The phases are separated by a moving interface. Moreover, the storage of foreign atoms in the crystals is accompanied by a volume change leading to mechanical stresses. The physical behavior of the storage process is described by a sharp-interface model.

The sharp-interface model consists of a system of parabolic and elliptic differential equations. The evolution of the interface is described by algebraic and ordinary differential equations, which are called jump conditions.

The regularization of the sharp-interface model by higher gradients and viscous terms leads to local phase field models. Here the jump of the concentration between the two phases is a smooth transition of thickness ε. Therefore the jump conditions at the interface are redundant. However, in the sharp limit ε to zero, the jump conditions of a sharp-interface model must be recovered.

By means of formal asymptotic methods several regularizations of the sharp-interface model are studied. One could show, that the regularization of the diffusion equation implies a regularization of the balance of momentum. Furthermore there are regularizations that lead to wrong jump conditions, so that they can not be used to describe the energy storage systems.


Publications

  Articles in Refereed Journals

  • 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, , 25 (2017) pp. 065015, DOI 10.1088/1361-651X/aa7862 .
    Abstract
    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. Heida, A. Mielke, Averaging of time-periodic dissipation potentials in rate-independent processes, Discrete and Continuous Dynamical Systems -- Series S, 10 (2017) pp. 1303--1327.
    Abstract
    We study the existence and well-posedness of rate-independent systems (or hysteresis operators) with a dissipation potential that oscillates in time with period ε. In particular, for the case of quadratic energies in a Hilbert space, we study the averaging limit ε→0 and show that the effctive dissipation potential is given by the minimum of all friction thresholds in one period, more precisely as the intersection of all the characteristic domains. We show that the rates of the process do not converge weakly, hence our analysis uses the notion of energetic solutions and relies on a detailed estimates to obtain a suitable equi-continuity of the solutions in the limit ε→0.

  • M. Heida, Stochastic homogenization of rate-independent systems, Continuum Mechanics and Thermodynamics, 29 (2017) pp. 853--894, DOI 10.1007/s00161-017-0564-z .
    Abstract
    We study the stochastic and periodic homogenization 1-homogeneous convex functionals. We proof some convergence results with respect to stochastic two-scale convergence, which are related to classical Gamma-convergence results. The main result is a general liminf-estimate for a sequence of 1-homogeneous functionals and a two-scale stability result for sequences of convex sets. We apply our results to the homogenization of rateindependent systems with 1-homogeneous dissipation potentials and quadratic energies. In these applications, both the energy and the dissipation potential have an underlying stochastic microscopic structure. We study the particular homogenization problems of Prandlt-Reuss plasticity, Coulomb friction on a macroscopic surface and Coulomb friction on microscopic fissure.

  • M. Liero, A. Mielke, M.A. Peletier, D.R.M. Renger, On microscopic origins of generalized gradient structures, Discrete and Continuous Dynamical Systems -- Series S, 10 (2017) pp. 1--35, DOI 10.3934/dcdss.2017001 .
    Abstract
    Classical gradient systems have a linear relation between rates and driving forces. In generalized gradient systems we allow for arbitrary relations derived from general non-quadratic dissipation potentials. This paper describes two natural origins for these structures. A first microscopic origin of generalized gradient structures is given by the theory of large-deviation principles. While Markovian diffusion processes lead to classical gradient structures, Poissonian jump processes give rise to cosh-type dissipation potentials. A second origin arises via a new form of convergence, that we call EDP-convergence. Even when starting with classical gradient systems, where the dissipation potential is a quadratic functional of the rate, we may obtain a generalized gradient system in the evolutionary Gamma-limit. As examples we treat (i) the limit of a diffusion equation having a thin layer of low diffusivity, which leads to a membrane model, and (ii) the limit of diffusion over a high barrier, which gives a reaction-diffusion system.

  • 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.
    Abstract
    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.

  • M. Heida, B. Schweizer, Non-periodic homogenization of infinitesimal strain plasticity equations, ZAMM. Zeitschrift für Angewandte Mathematik und Mechanik, 96 (2016) pp. 5--23.

  • M. Khodayari, P. Reinsberg, A.A. Abd-El-Latif, Ch. Merdon, J. Fuhrmann, H. Baltruschat, Determining solubility and diffusivity by using a flow cell coupled to a mass spectrometer, ChemPhysChem, 17 (2016) pp. 1647--1655.

  • W. Dreyer, C. Guhlke, R. Müller, A new perspective on the electron transfer: Recovering the Butler--Volmer equation in non-equilibrium thermodynamics, Physical Chemistry Chemical Physics, 18 (2016) pp. 24966--24983.
    Abstract
    Understanding and correct mathematical description of electron transfer reaction is a central question in electrochemistry. Typically the electron transfer reactions are described by the Butler-Volmer equation which has its origin in kinetic theories. The Butler-Volmer equation relates interfacial reaction rates to bulk quantities like the electrostatic potential and electrolyte concentrations. Since in the classical form, the validity of the Butler-Volmer equation is limited to some simple electrochemical systems, many attempts have been made to generalize the Butler-Volmer equation. Based on non-equilibrium thermodynamics we have recently derived a reduced model for the electrode-electrolyte interface. This reduced model includes surface reactions but does not resolve the charge layer at the interface. Instead it is locally electroneutral and consistently incorporates all features of the double layer into a set of interface conditions. In the context of this reduced model we are able to derive a general Butler-Volmer equation. We discuss the application of the new Butler-Volmer equations to different scenarios like electron transfer reactions at metal electrodes, the intercalation process in lithium-iron-phosphate electrodes and adsorption processes. We illustrate the theory by an example of electroplating.

  • K. Disser, M. Liero, On gradient structures for Markov chains and the passage to Wasserstein gradient flows, Networks and Heterogeneous Media, 10 (2015) pp. 233-253.
    Abstract
    We study the approximation of Wasserstein gradient structures by their finite-dimensional analog. We show that simple finite-volume discretizations of the linear Fokker-Planck equation exhibit the recently established entropic gradient-flow structure for reversible Markov chains. Then, we reprove the convergence of the discrete scheme in the limit of vanishing mesh size using only the involved gradient-flow structures. In particular, we make no use of the linearity of the equations nor of the fact that the Fokker-Planck equation is of second order.

  • D. Peschka, Thin-film free boundary problems for partial wetting, Journal of Computational Physics, 295 (2015) pp. 770--778.
    Abstract
    We present a novel framework to solve thin-film equations with an explicit non-zero contact angle, where the support of the solution is treated as an unknown. The algorithm uses a finite element method based on a gradient formulation of the thin-film equations coupled to an arbitrary Lagrangian-Eulerian method for the motion of the support. Features of this algorithm are its simplicity and robustness. We apply this algorithm in 1D and 2D to problems with surface tension, contact angles and with gravity.

  • P. Colli, G. Gilardi, E. Rocca, J. Sprekels, Vanishing viscosities and error estimate for a Cahn--Hilliard type phase field system related to tumor growth, Nonlinear Analysis. Real World Applications. An International Multidisciplinary Journal, 26 (2015) pp. 93--108.
    Abstract
    In this paper we perform an asymptotic analysis for two different vanishing viscosity coefficients occurring in a phase field system of Cahn--Hilliard type that was recently introduced in order to approximate a tumor growth model. In particular, we extend some recent results obtained in [Colli-Gilardi-Hilhorst 2015], letting the two positive viscosity parameters tend to zero independently from each other and weakening the conditions on the initial data in such a way as to maintain the nonlinearities of the PDE system as general as possible. Finally, under proper growth conditions on the interaction potential, we prove an error estimate leading also to the uniqueness result for the limit system.

  • M.G. Hennessy, V.M. Burlakov, A. Münch, B. Wagner, A. Goriely, Controlled topological transitions in thin-film phase separation, SIAM Journal on Applied Mathematics, 75 (2015) pp. 38--60.
    Abstract
    In this paper the evolution of a binary mixture in a thin-film geometry with a wall at the top and bottom is considered. Bringing the mixture into its miscibility gap so that no spinodal decomposition occurs in the bulk, a slight energetic bias of the walls towards each one of the constituents ensures the nucleation of thin boundary layers that grow until the constituents have moved into one of the two layers. These layers are separated by an interfacial region, where the composition changes rapidly. Conditions that ensure the separation into two layers with a thin interfacial region are investigated based on a phase-field model and using matched asymptotic expansions a corresponding sharp-interface problem for the location of the interface is established. It is then argued that a thus created two-layer system is not at its energetic minimum but destabilizes into a controlled self-replicating pattern of trapezoidal vertical stripes by minimizing the interfacial energy between the phases while conserving their area. A quantitative analysis of this mechanism is carried out via a new thin-film model for the free interfaces, which is derived asymptotically from the sharp-interface model.

  • 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.
    Abstract
    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.

  • CH. Bayer, H.A. Hoel, A. Kadir, P. Plechac, M. Sandberg, A. Szepessy, Computational error estimates for Born--Oppenheimer molecular dynamics with nearly crossing potential surfaces, Applied Mathematics Research Express, 2015 (2015) pp. 329--417.

  • A. Caiazzo, I. Ramis-Conde, Multiscale modeling of palisade formation in glioblastoma multiforme, Journal of Theoretical Biology, 383 (2015) pp. 145--156.
    Abstract
    Palisades are characteristic tissue aberrations that arise in glioblastomas. Observation of palisades is considered as a clinical indicator of the transition from a noninvasive to an invasive tumour. In this article we propose a computational model to study the influence of genotypic and phenotypic heterogeneity in palisade formation. For this we produced three dimensional realistic simulations, based on a multiscale hybrid model, coupling the evolution of tumour cells and the oxygen diffusion in tissue, that depict the shape of palisades during its formation. Our results can be summarized as the following: (1) we show that cell heterogeneity is a crucial factor in palisade formation and tumour growth; (2) we present results that can explain the observed fact that recursive tumours are more malignant than primary tumours; and (3) the presented simulations can provide to clinicians and biologists for a better understanding of palisades 3D structure as well as glioblastomas growth dynamics

  • W. Dreyer, C. Guhlke, R. Müller, Modeling of electrochemical double layers in thermodynamic non-equilibrium, Physical Chemistry Chemical Physics, 17 (2015) pp. 27176--27194.
    Abstract
    We consider the contact between an electrolyte and a solid electrode. At first we formulate a thermodynamic consistent model that resolves boundary layers at interfaces. The model includes charge transport, diffusion, chemical reactions, viscosity, elasticity and polarization under isothermal conditions. There is a coupling between these phenomena that particularly involves the local pressure in the electrolyte. Therefore the momentum balance is of major importance for the correct description of the layers.

    The width of the boundary layers is typically very small compared to the macroscopic dimensions of the system. In a second step we thus apply the method of asymptotic analysis to derive a simpler reduced model that does not resolve the boundary layers but instead incorporates the electrochemical properties of the layers into a set of new boundary conditions. For a metal-electrolyte interface, we derive a qualitative description of the double layer capacitance without the need to resolve space charge layers.

  • H. Hanke, D. Knees, Homogenization of elliptic systems with non-periodic, state dependent coefficients, Asymptotic Analysis, 92 (2015) pp. 203--234.
    Abstract
    In this paper, a homogenization problem for an elliptic system with non-periodic, state dependent coefficients representing microstructure is investigated. The state functions defining the tensor of coefficients are assumed to have an intrinsic length scale denoted by ε > 0. The aim is the derivation of an effective model by investigating the limit process ε → 0 of the state functions rigorously. The effective model is independent of the parameter ε > 0 but preserves the microscopic structure of the state functions (ε > 0), meaning that the effective tensor is given by a unit cell problem prescribed by a suitable microscopic tensor. Due to the non-periodic structure of the state functions and the corresponding microstructure, the effective tensor turns out to vary from point to point (in contrast to a periodic microscopic model). In a forthcoming paper, these states will be solutions of an additional evolution law describing changes of the microstructure. Such changes could be the consequences of temperature changes, phase separation or damage progression, for instance. Here, in addition to the above and as a preparation for an application to time-dependent damage models (discussed in a future paper), we provide a Γ-convergence result of sequences of functionals being related to the previous microscopic models with state dependent coefficients. This requires a penalization term for piecewise constant state functions that allows us to extract from bounded sequences those sequences converging to a Sobolev function in some sense. The construction of the penalization term is inspired by techniques for Discontinuous Galerkin methods and is of own interest. A compactness and a density result are provided.

  • S. Jachalski, G. Kitavtsev, R. Taranets, Weak solutions to lubrication systems describing the evolution of bilayer thin films, Communications in Mathematical Sciences, 12 (2014) pp. 527--544.
    Abstract
    The existence of global nonnegative weak solutions is proved for coupled one-dimen- sional lubrication systems that describe the evolution of nanoscopic bilayer thin polymer films that take account of Navier-slip or no-slip conditions at both liquid-liquid and liquid-solid interfaces. In addition, in the presence of attractive van der Waals and repulsive Born intermolecular interactions existence of positive smooth solutions is shown.

  • G. Aki, W. Dreyer, J. Giesselmann, Ch. Kraus, A quasi-incompressible diffuse interface model with phase transition, Mathematical Models & Methods in Applied Sciences, 24 (2014) pp. 827--861.
    Abstract
    This work introduces a new thermodynamically consistent diffuse model for two-component flows of incompressible fluids. For the introduced diffuse interface model, we investigate physically admissible sharp interface limits by matched asymptotic techniques. To this end, we consider two scaling regimes where in one case we recover the Euler equations and in the other case the Navier-Stokes equations in the bulk phases equipped with admissible interfacial conditions. For the Navier-Stokes regime, we further assume the densities of the fluids are close to each other in the sense of a small parameter which is related to the interfacial thickness of the diffuse model.

  • P. Colli, G. Gilardi, P. Krejčí, J. Sprekels, A vanishing diffusion limit in a nonstandard system of phase field equations, Evolution Equations and Control Theory, 3 (2014) pp. 257--275.
    Abstract
    We are concerned with a nonstandard phase field model of Cahn--Hilliard type. The model, which was introduced by Podio-Guidugli (Ric. Mat. 2006), describes two-species phase segregation and consists of a system of two highly nonlinearly coupled PDEs. It has been recently investigated by Colli, Gilardi, Podio-Guidugli, and Sprekels in a series of papers: see, in particular, SIAM J. Appl. Math. 2011, and Boll. Unione Mat. Ital. 2012. In the latter contribution, the authors can treat the very general case in which the diffusivity coefficient of the parabolic PDE is allowed to depend nonlinearly on both variables. In the same framework, this paper investigates the asymptotic limit of the solutions to the initial-boundary value problems as the diffusion coefficient $sigma$ in the equation governing the evolution of the order parameter tends to zero. We prove that such a limit actually exists and solves the limit problem, which couples a nonlinear PDE of parabolic type with an ODE accounting for the phase dynamics. In the case of a constant diffusivity, we are able to show uniqueness and to improve the regularity of the solution.

  • A. Caiazzo, J. Mura, Multiscale modeling of weakly compressible elastic materials in harmonic regime and application to microscale structure estimation, Multiscale Modeling & Simulation. A SIAM Interdisciplinary Journal, 12 (2014) pp. 514--537.
    Abstract
    This article is devoted to the modeling of elastic materials composed by an incompressible elastic matrix and small compressible gaseous inclusions, under a time harmonic excitation. In a biomedical context, this model describes the dynamics of a biological tissue (e.g. lung or liver) when wave analysis methods (such as Magnetic Resonance Elastography) are used to estimate tissue properties. Due to the multiscale nature of the problem, direct numerical simulations are prohibitive. We extend the homogenized model introduced in [Baffico, Grandmont, Maday, Osses, SIAM J. Mult. Mod. Sim., 7(1), 2008] to a time harmonic regime to describe the solid-gas mixture from a macroscopic point of view in terms of an effective elasticity tensor. Furthermore, we derive and validate numerically analytical approximations for the effective elastic coefficients in terms of macroscopic parameters. This simplified description is used to to set up an efficient variational approach for the estimation of the tissue porosity, using the mechanical response to external harmonic excitations.

  • W. Dreyer, J. Giesselmann, Ch. Kraus, A compressible mixture model with phase transition, Physica D. Nonlinear Phenomena, 273--274 (2014) pp. 1--13.
    Abstract
    We introduce a new thermodynamically consistent diffuse interface model of Allen-Cahn/Navier-Stokes type for multi-component flows with phase transitions and chemical reactions. For the introduced diffuse interface model, we investigate physically admissible sharp interface limits by matched asymptotic techniques. We consider two scaling regimes, i.e. a non-dissipative and a dissipative regime, where we recover in the sharp interface limit a generalized Allen-Cahn/Euler system for mixtures with chemical reactions in the bulk phases equipped with admissible interfacial conditions. The interfacial conditions satify, for instance, a Young-Laplace and a Stefan type law.

  • S. Neukamm, H. Olbermann, Homogenization of the nonlinear bending theory for plates, Calculus of Variations and Partial Differential Equations, (published online on Sept. 14, 2014) pp. , DOI 10.1007/s00526-014-0765-2 .
    Abstract
    We carry out the spatially periodic homogenization of nonlinear bending theory for plates. The derivation is rigorous in the sense of Gamma-convergence. In contrast to what one naturally would expect, our result shows that the limiting functional is not simply a quadratic functional of the second fundamental form of the deformed plate as it is the case in nonlinear plate theory. It turns out that the limiting functional discriminates between whether the deformed plate is locally shaped like a "cylinder" or not. For the derivation we investigate the oscillatory behavior of sequences of second fundamental forms associated with isometric immersions, using two-scale convergence. This is a non-trivial task, since one has to treat two-scale convergence in connection with a nonlinear differential constraint.

  • P. Colli, G. Gilardi, P. Podio-Guidugli, J. Sprekels, An asymptotic analysis for a nonstandard Cahn--Hilliard system with viscosity, Discrete and Continuous Dynamical Systems -- Series S, 6 (2013) pp. 353--368.
    Abstract
    This paper is concerned with a diffusion model of phase-field type, consisting of a parabolic system of two partial differential equations, interpreted as balances of microforces and microenergy, for two unknowns: the problem's order parameter $rho$ and the chemical potential $mu$; each equation includes a viscosity term -- respectively, $varepsilon,partial_tmu$ and $delta,partial_trho$ -- with $varepsilon$ and $delta$ two positive parameters; the field equations are complemented by Neumann homogeneous boundary conditions and suitable initial conditions. In a recent paper [5], we proved that this problem is well-posed and investigated the long-time behavior of its $(varepsilon,delta)-$solutions. Here we discuss the asymptotic limit of the system as $eps$ tends to 0. We prove convergence of $(varepsilon,delta)-$solutions to the corresponding solutions for the case $eps$ =0, whose long-time behavior we characterize; in the proofs, we employ compactness and monotonicity arguments.

  • P. Colli, G. Gilardi, P. Podio-Guidugli, J. Sprekels, Global existence and uniqueness for a singular/degenerate Cahn--Hilliard system with viscosity, Journal of Differential Equations, 254 (2013) pp. 4217--4244.
    Abstract
    Existence and uniqueness are investigated for a nonlinear diffusion problem of phase-field type, consisting of a parabolic system of two partial differential equations, complemented by Neumann homogeneous boundary conditions and initial conditions. This system aims to model two-species phase segregation on an atomic [19]; in the balance equations of microforces and microenergy, the two unknowns are the order parameter $rho$ and the chemical potential $mu$. A simpler version of the same system has recently been discussed in [8]. In this paper, a fairly more general phase-field equation for $rho$ is coupled with a genuinely nonlinear diffusion equation for $mu$. The existence of a global-in-time solution is proved with the help of suitable a priori estimates. In the case of costant atom mobility, a new and rather unusual uniqueness proof is given, based on a suitable combination of variables.

  • P. Colli, G. Gilardi, P. Podio-Guidugli, J. Sprekels, Continuous dependence for a nonstandard Cahn--Hilliard system with nonlinear atom mobility, Rendiconti del Seminario Matematico. Universita e Politecnico Torino, 70 (2012) pp. 27--52.
    Abstract
    This note is concerned with a nonlinear diffusion problem of phase-field type, consisting of a parabolic system of two partial differential equations, complemented by Neumann homogeneous boundary conditions and initial conditions. The system arises from a model of two-species phase segregation on an atomic lattice [Podio-Guidugli 2006]; it consists of the balance equations of microforces and microenergy; the two unknowns are the order parameter $rho$ and the chemical potential $mu$. Some recent results obtained for this class of problems is reviewed and, in the case of a nonconstant and nonlinear atom mobility, uniqueness and continuous dependence on the initial data are shown with the help of a new line of argumentation developed in Colli/Gilardi/Podio-Guidugli/Sprekels 2012.

  • P. Colli, G. Gilardi, P. Podio-Guidugli, J. Sprekels, Global existence for a strongly coupled Cahn--Hilliard system with viscosity, Bollettino della Unione Matematica Italiana. Serie 9, 5 (2012) pp. 495--513.
    Abstract
    An existence result is proved for a nonlinear diffusion problem of phase-field type, consisting of a parabolic system of two partial differential equations, complemented by Neumann homogeneous boundary conditions and initial conditions. This system is meant to model two-species phase segregation on an atomic lattice under the presence of diffusion. A similar system has been recently introduced and analyzed in [CGPS11]. Both systems conform to the general theory developed in [Pod06]: two parabolic PDEs, interpreted as balances of microforces and microenergy, are to be solved for the order parameter $rho$ and the chemical potential $mu$. In the system studied in this note, a phase-field equation in $rho$ fairly more general than in [CGPS11] is coupled with a highly nonlinear diffusion equation for $mu$, in which the conductivity coefficient is allowed to depend nonlinearly on both variables.

  • G. Aki, J. Daube, W. Dreyer, J. Giesselmann, M. Kränkel, Ch. Kraus, A diffuse interface model for quasi-incompressible flows: Sharp interface limits and numerics, ESAIM Proceedings, 38 (2012) pp. 54--77.
    Abstract
    In this contribution, we investigate a diffuse interface model for quasi-incompressible flows. We determine corresponding sharp interface limits of two different scalings. The sharp interface limit is deduced by matched asymptotic expansions of the fields in powers of the interface. In particular, we study solutions of the derived system of inner equations and discuss the results within the general setting of jump conditions for sharp interface models. Furthermore, we treat, as a subproblem, the convective Cahn-Hilliard equation numerically by a Local Discontinuous Galerkin scheme.

  • W. Dreyer, J. Giesselmann, Ch. Kraus, Ch. Rohde, Asymptotic analysis for Korteweg models, Interfaces and Free Boundaries. Mathematical Modelling, Analysis and Computation, 14 (2012) pp. 105--143.
    Abstract
    This paper deals with a sharp interface limit of the isothermal Navier-Stokes-Korteweg system. The sharp interface limit is performed by matched asymptotic expansions of the fields in powers of the interface width. These expansions are considered in the interfacial region (inner expansions) and in the bulk (outer expansion) and are matched order by order. Particularly we consider the first orders of the corresponding inner equations obtained by a change of coordinates in an interfacial layer. For a specific scaling we establish solvability criteria for these inner equations and recover the results within the general setting of jump conditions for sharp interface models.

  • K. Hermsdörfer, Ch. Kraus, D. Kröner, Interface conditions for limits of the Navier--Stokes--Korteweg model, Interfaces and Free Boundaries. Mathematical Modelling, Analysis and Computation, 13 (2011) pp. 239--254.
    Abstract
    In this contribution we will study the behaviour of the pressure across phase boundaries in liquid-vapour flows. As mathematical model we will consider the static version of the Navier-Stokes-Korteweg model which belongs to the class of diffuse interface models. From this static equation a formula for the pressure jump across the phase interface can be derived. If we perform then the sharp interface limit we see that the resulting interface condition for the pressure seems to be inconsistent with classical results of hydrodynamics. Therefore we will present two approaches to recover the results of hydrodynamics in the sharp interface limit at least for special situations.

  • G. Kitavtsev, L. Recke, B. Wagner, Center manifold reduction approach for the lubrication equation, Nonlinearity, 24 (2011) pp. 2347--2369.
    Abstract
    The goal of this study is the reduction of the lubrication equation, modelling thin film dynamics, onto an approximate invariant manifold. The reduction is derived for the physical situation of the late phase evolution of a dewetting thin liquid film, where arrays of droplets connected by an ultrathin film of thickness $eps$ undergo a slow-time coarsening dynamics. With this situation in mind, we construct an asymptotic approximation of the corresponding invariant manifold, that is parametrized by a family of droplet pressures and positions, in the limit when $epsto 0$. The approach is inspired by the paper by Mielke and Zelik [Mem. Amer. Math. Soc., Vol. 198, 2009], where the center manifold reduction was carried out for a class of semilinear systems. In this study this approach is considered for quasilinear degenerate parabolic PDE's such as lubrication equations. While it has previously been shown by Glasner and Witelski [Phys. Rev. E, Vol. 67, 2003], that the system of ODEs governing the coarsening dynamics, can be obtained via formal asymptotic methods, the center manifold reduction approach presented here, pursues the rigorous justification of this asymptotic limit.

  • A. Münch, B. Wagner, Impact of slippage on the morphology and stability of a dewetting rim, Journal of Physics: Condensed Matter, 23 (2011) pp. 184101/1--184101/12.
    Abstract
    In this study lubrication theory is used to describe the stability and morphology of the rim that forms as a thin polymer film dewets from a hydrophobized silicon wafer. Thin film equations are derived from the governing hydrodynamic equations for the polymer to enable the systematic mathematical and numerical analysis of the properties of the solutions for different regimes of slippage and for a range of time scales. Dewetting rates and the cross sectional profiles of the evolving rims are derived for these models and compared to experimental results. Experiments also show that the rim is typically unstable in the spanwise direction and develops thicker and thinner parts that may grow into “fingers”. Linear stability analysis as well as nonlinear numerical solutions are presented to investigate shape and growth rate of the rim instability. It is demonstrated that the difference in morphology and the rate at which the instability develops can be directly attributed to the magnitude of slippage. Finally, a derivation is given for the dominant wavelength of the bulges along the unstable rim.

  • A. Münch, C.P. Please, B. Wagner, Spin coating of an evaporating polymer solution, Physics of Fluids, 23 (2011) pp. 102101/1--102101/12.
    Abstract
    We consider a mathematical model of spin coating of a single polymer blended in a solvent. The model describes the one-dimensional development of the thin layer of the mixture as the layer thins due to flow created by a balance of viscous forces and centrifugal forces and due to evaporation of the solvent. In the model both the diffusivity of the solvent in the polymer and the viscosity of the mixture are very rapidly varying functions of the solvent volume fraction. Guided by numerical solutions an asymptotic analysis reveals a number of different possible behaviours of the thinning layer dependent on the nondimensional parameters describing the system. The main practical interest is in controlling the appearance and development of a “skin” on the polymer where the solvent concentration reduces rapidly on the outer surface leaving the bulk of the layer still with high concentrations of solvent. The critical parameters controlling this behaviour are found to be $eps$ the ratio of the diffusion to advection time scales, $delta$ the ratio of the evaporation to advection time scales and $exp(-gamma)$, the ratio of the diffusivity of the initial mixture and the pure polymer. In particular, our analysis shows that for very small evaporation with $delta ll exp(-3/(4gamma)) eps^3/4$ skin formation can be prevented.

  • G. Kitavtsev, B. Wagner, Coarsening dynamics of slipping droplets, Journal of Engineering Mathematics, 66 (2010) pp. 271--292.
    Abstract
    This paper studies the late phase dewetting process of nanoscopic thin polymer films on hydrophobized substrates using some recently derived lubrication models that take account of large slippage at the polymer-substrate interface. The late phase of this process is characterized by the slow-time coarsening dynamics of arrays of droplets that remain after rupture and the initial dewetting phases. For this situation a reduced system of ordinary differential equations is derived from the lubrication model for large slippage using asymptotic analysis. This extends known results for the no-slip case. On the basis of the reduced model, the role of the slippage as a control parameter for droplet migration is analysed and several new qualitative effects for the coarsening process are identified.

  • D. Peschka, A. Münch, B. Niethammer, Self-similar rupture of viscous thin films in the strong-slip regime, Nonlinearity, 23 (2010) pp. 409--427.

  • D. Peschka, A. Münch, B. Niethammer, Thin film rupture for large slip, Journal of Engineering Mathematics, 66 (2010) pp. 33--51.
    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 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.

  • W. Dreyer, Ch. Kraus, On the van der Waals--Cahn--Hilliard phase-field model and its equilibria conditions in the sharp interface limit, Proceedings of the Royal Society of Edinburgh. Section A. Mathematics, 140 A (2010) pp. 1161--1186.
    Abstract
    We study the equilibria of liquid--vapor phase transitions of a single substance at constant temperature and relate the sharp interface model of classical thermodynamics to a phase field model that determines the equilibria by the stationary van der Waals--Cahn--Hilliard theory.
    For two reasons we reconsider this old problem. 1. Equilibria in a two phase system can be established either under fixed total volume of the system or under fixed external pressure. The latter case implies that the domain of the two--phase system varies. However, in the mathematical literature rigorous sharp interface limits of phase transitions are usually considered under fixed volume. This brings the necessity to extend the existing tools for rigorous sharp interface limits to changing domains since in nature most processes involving phase transitions run at constant pressure. 2. Thermodynamics provides for a single substance two jump conditions at the sharp interface, viz. the continuity of the specific Gibbs free energies of the adjacent phases and the discontinuity of the corresponding pressures, which is balanced by the mean curvature. The existing estimates for rigorous sharp interface limits show only the first condition. We identify the cause of this phenomenon and develop a strategy that yields both conditions up to the first order.
    The necessary information on the equilibrium conditions are achieved by an asymptotic expansion of the density which is valid for an arbitrary double well potential. We establish this expansion by means of local energy estimates, uniform convergence results of the density and estimates on the Laplacian of the density.

  • D. Peschka, A. Münch, B. Niethammer, Self-similar rupture of viscous thin films in the strong-slip regime, Nonlinearity, 23 (2010) pp. 409--427.
    Abstract
    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.

  • J.R. King, A. Münch, B. Wagner, Linear stability analysis of a sharp-interface model for dewetting thin films, Journal of Engineering Mathematics, 63 (2009) pp. 177--195.
    Abstract
    The topic of this study concerns the stability of the three-phase contact-line of a dewetting thin liquid film on a hydrophobised substrate driven by van der Waals forces. The role of slippage in the emerging instability at the three-phase contact-line is studied by deriving a sharp-interface model for the dewetting thin film via matched asymptotic expansions. This allows for a derivation of travelling waves and their linear stability via eigenmode analysis. In contrast to the dispersion relations typically encountered for the finger-instabilty, where the dependence of the growth rate on the wave number is quadratic, here it is linear. Using the separation of time scales of the slowly growing rim of the dewetting film and time scale on which the contact line destabilises, the sharp-interface results are compared to earlier results for the full lubrication model and good agreement for the most unstable modes is obtained.

  • K. Afanasiev, A. Münch, B. Wagner, Thin film dynamics on a vertically rotating disk partially immersed in a liquid bath, Applied Mathematical Modelling. Simulation and Computation for Engineering and Environmental Systems. Elsevier Science Inc., New York, NY. English, English abstracts., 32 (2008) pp. 1894-1911.
    Abstract
    The axisymmetric flow of a thin liquid film subject to surface tension, gravity and centrifugal forces is considered for the problem of a vertically rotating disk that is partially immersed in a liquid bath. This problem constitutes a generalization of the classic Landau-Levich drag-out problem to axisymmetric flow. A generalized lubrication model that includes the meniscus region connecting the thin film to the bath is derived. The resulting nonlinear fourth-order partial differential equation is solved numerically using a finite element scheme. For a range of parameters steady states are found. While the solutions for the height profile of the film near the drag-out region show excellent agreement with the asymptotic solutions to the corresponding classic Landau-Levich problem, they show novel patterns away from the meniscus region. The implications for possible industrial applications are discussed.

  • A. Münch, B. Wagner, Galerkin method for feedback controlled Rayleigh--Bénard convection, Nonlinearity, 21 (2008) pp. 2625-2651.

  • 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.

  • K. Afanasiev, A. Münch, B. Wagner, On the Landau--Levich problem for non-Newtonian liquids, Physical Review E. Statistical, Nonlinear, and Soft Matter Physics, 76 (2007) pp. 036307/1--036307/12.
    Abstract
    In this paper the drag-out problem for shear-thinning liquids at variable inclination angle is considered. For this free boundary problem dimension-reduced lubrication equations are derived for the most commonly used viscosity models, namely, the power-law, Ellis and Carreau model. For the resulting lubrication models a system of ordinary differential equation governing the steady state solutions is obtained. Phase plane analysis is used to characterize the type of possible steady state solutions and their dependence on the rheological parameters.

  • P. Krejčí, J. Sprekels, Elastic-ideally plastic beams and Prandtl--Ishlinskii hysteresis operators, Mathematical Methods in the Applied Sciences, 30 (2007) pp. 2371--2393.

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

  • P.L. Evans, J.R. King, A. Münch, Intermediate-asymptotic structure of a dewetting rim with strong slip, Applied Mathematics Research Express, (2006) pp. 25262/1--25262/25.
    Abstract
    When a thin viscous liquid film dewets, it typically forms a rim which spreads outwards, leaving behind a growing dry region. We consider the dewetting behaviour of a film, when there is strong slip at a liquid-substrate interface. The film can be modelled by two coupled partial differential equations (PDEs) describing the film thickness and velocity. Using asymptotic methods, we describe the structure of the rim as it evolves in time, and the rate of dewetting, in the limit of large slip lengths. An inner region emerges, closest to the dewetted region, where surface tension is important; in an outer region, three subregions develop. This asymptotic description is compared with numerical solutions of the full system of PDEs.

  • J.R. King, A. Münch, B. Wagner, Linear stability of a ridge, Nonlinearity, 19 (2006) pp. 2813-2831.
    Abstract
    We investigate the stability of the three-phase contact-line of a thin liquid ridge on a hydrophobic substrate for flow driven by surface tension and van der Waals forces. We study the role of slippage in the emerging instability at the three-phase contact-line by comparing the lubrication models for no-slip and slip-dominated conditions at the liquid/substrate interface. For both cases we derive a sharp-interface model via matched asymptotic expansions and derive the eigenvalues from a linear stability analysis of the respective reduced models. We compare our asymptotic results with the eigenvalues obtained numerically for the full lubrication models.

  • A. Münch, B. Wagner, M. Rauscher, R. Blossey, A thin film model for corotational Jeffreys fluids under strong slip, The European Physical Journal. E. Soft Matter, 20 (2006) pp. 365-368.
    Abstract
    We derive a thin film model for viscoelastic liquids under strong slip which obey the stress tensor dynamics of corotational Jeffreys fluids.

  • B. Jin, A. Acrivos, A. Münch, The drag-out problem in film coating, Physics of Fluids, 17 (2005) pp. 103603/1-103603/12.

  • A. Münch, P.L. Evans, Marangoni-driven liquid films rising out of a meniscus onto a nearly-horizontal substrate, Physica D. Nonlinear Phenomena, 209 (2005) pp. 164--177.

  • A. Münch, B. Wagner, Th.P. Witelski, Lubrication models for small to large slip lengths, Journal of Engineering Mathematics, 53 (2005) pp. 359-383.

  • A. Münch, B. Wagner, Contact-line instability for dewetting thin films, Physica D. Nonlinear Phenomena, 209 (2005) pp. 178--190.

  Preprints, Reports, Technical Reports

  • L. Donati, M. Heida, M. Weber, B. Keller, Estimation of the infinitesimal generator by square-root approximation, Preprint no. 2416, WIAS, Berlin, 2017, DOI 10.20347/WIAS.PREPRINT.2416 .
    Abstract, PDF (1739 kByte)
    For the analysis of molecular processes, the estimation of time-scales, i.e., transition rates, is very important. Estimating the transition rates between molecular conformations is -- from a mathematical point of view -- an invariant subspace projection problem. A certain infinitesimal generator acting on function space is projected to a low-dimensional rate matrix. This projection can be performed in two steps. First, the infinitesimal generator is discretized, then the invariant subspace is approximated and used for the subspace projection. In our approach, the discretization will be based on a Voronoi tessellation of the conformational space. We will show that the discretized infinitesimal generator can simply be approximated by the geometric average of the Boltzmann weights of the Voronoi cells. Thus, there is a direct correlation between the potential energy surface of molecular structures and the transition rates of conformational changes. We present results for a 2d-diffusion process and Alanine dipeptide.

  • M. Heida, On convergences of the squareroot approximation scheme to the Fokker--Planck operator, Preprint no. 2399, WIAS, Berlin, 2017, DOI 10.20347/WIAS.PREPRINT.2399 .
    Abstract, PDF (506 kByte)
    We study the qualitative convergence properties of a finite volume scheme that recently was proposed by Lie, Fackeldey and Weber [SIAM Journal on Matrix Analysis and Applications 2013 (34/2)] in the context of conformation dynamics. The scheme was derived from physical principles and is called the squareroot approximation (SQRA) scheme. We show that solutions to the SQRA equation converge to solutions of the Fokker-Planck equation using a discrete notion of G-convergence. Hence the squareroot approximation turns out to be a usefull approximation scheme to the Fokker-Planck equation in high dimensional spaces. As an example, in the special case of stationary Voronoi tessellations we use stochastic two-scale convergence to prove that this setting satisfies the G-convergence property. In particular, the class of tessellations for which the G-convergence result holds is not trivial.

  • E. Meca Álvarez, A. Münch, B. Wagner, Localized instabilities and spinodal decomposition in driven systems in the presence of elasticity, Preprint no. 2387, WIAS, Berlin, 2017, DOI 10.20347/WIAS.PREPRINT.2387 .
    Abstract, PDF (801 kByte)
    We study numerically and analytically the instabilities associated with phase separation in a solid layer on which an external material flux is imposed. The first instability is localized within a boundary layer at the exposed free surface by a process akin to spinodal decomposition. In the limiting static case, when there is no material flux, the coherent spinodal decomposition is recovered. In the present problem stability analysis of the time-dependent and non-uniform base states as well as numerical simulations of the full governing equations are used to establish the dependence of the wavelength and onset of the instability on parameter settings and its transient nature as the patterns eventually coarsen into a flat moving front. The second instability is related to the Mullins-Sekerka instability in the presence of elasticity and arises at the moving front between the two phases when the flux is reversed. Stability analyses of the full model and the corresponding sharp-interface model are carried out and compared. Our results demonstrate how interface and bulk instabilities can be analysed within the same framework which allows to identify and distinguish each of them clearly. The relevance for a detailed understanding of both instabilities and their interconnections in a realistic setting are demonstrated for a system of equations modelling the lithiation/delithiation processes within the context of Lithium ion batteries.

  • M. Heida, S. Nesenenko, Stochastic homogenization of rate-dependent models of monotone type in plasticity, Preprint no. 2366, WIAS, Berlin, 2017, DOI 10.20347/WIAS.PREPRINT.2366 .
    Abstract, PDF (548 kByte)
    In this work we deal with the stochastic homogenization of the initial boundary value problems of monotone type. The models of monotone type under consideration describe the deformation behaviour of inelastic materials with a microstructure which can be characterised by random measures. Based on the Fitzpatrick function concept we reduce the study of the asymptotic behaviour of monotone operators associated with our models to the problem of the stochastic homogenization of convex functionals within an ergodic and stationary setting. The concept of Fitzpatrick's function helps us to introduce and show the existence of the weak solutions for rate-dependent systems. The derivations of the homogenization results presented in this work are based on the stochastic two-scale convergence in Sobolev spaces. For completeness, we also present some two-scale homogenization results for convex functionals, which are related to the classical Gamma-convergence theory.

  • E. Meca Álvarez, A. Münch, B. Wagner, Sharp-interface formation during lithium intercalation into silicon, Preprint no. 2257, WIAS, Berlin, 2016.
    Abstract, PDF (518 kByte)
    In this study we present a phase-field model that describes the process of intercalation of Li ions into a layer of an amorphous solid such as a-Si. The governing equations couple a viscous Cahn-Hilliard-Reaction model with elasticity in the framework of the Cahn-Larché system. We discuss the parameter settings and flux conditions at the free boundary that lead to the formation of phase boundaries having a sharp gradient in ion concentration between the initial state of the solid layer and the intercalated region. We carry out a matched asymptotic analysis to derive the corresponding sharp-interface model that also takes into account the dynamics of triple points where the sharp interface in the bulk of the layer intersects the free boundary. We numerically compare the interface motion predicted by the sharp-interface model with the long-time dynamics of the phase-field model.

  • A. Münch, B. Wagner, L.P. Cook, R.R. Braun, Apparent slip for an upper convected Maxwell fluid, Preprint no. 2255, WIAS, Berlin, 2016.
    Abstract, PDF (367 kByte)
    In this study the flow field of a nonlocal, diffusive upper convected Maxwell (UCM) fluid with a polymer in a solvent undergoing shearing motion is investigated for pressure driven planar channel flow and the free boundary problem of a liquid layer on a solid substrate. For large ratios of the zero shear polymer viscosity to the solvent viscosity, it is shown that channel flows exhibit boundary layers at the channel walls. In addition, for increasing stress diffusion the flow field away from the boundary layers undergoes a transition from a parabolic to a plug flow. Using experimental data for the wormlike micelle solutions CTAB/NaSal and CPyCl/NaSal, it is shown that the analytic solution of the governing equations predicts these signatures of the velocity profiles. Corresponding flow structures and transitions are found for the free boundary problem of a thin layer sheared along a solid substrate. Matched asymptotic expansions are used to first derive sharp-interface models describing the bulk flow with expressions for an em apparent slip for the boundary conditions, obtained by matching to the flow in the boundary layers. For a thin film geometry several asymptotic regimes are identified in terms of the order of magnitude of the stress diffusion, and corresponding new thin film models with a slip boundary condition are derived.

  • A. Caiazzo, F. Caforio, G. Montecinos, L.O. Müller, P.J. Blanco, E.F. Toro, Assessment of reduced order Kalman filter for parameter identification in one-dimensional blood flow models using experimental data, Preprint no. 2248, WIAS, Berlin, 2016.
    Abstract, PDF (8646 kByte)
    This work presents a detailed investigation of a parameter estimation approach based on the reduced order unscented Kalman filter (ROUKF) in the context of one-dimensional blood flow models. In particular, the main aims of this study are (i) to investigate the effect of using real measurements vs. synthetic data (i.e., numerical results of the same in silico model, perturbed with white noise) for the estimation and (ii) to identify potential difficulties and limitations of the approach in clinically realistic applications in order to assess the applicability of the filter to such setups. For these purposes, our numerical study is based on the in vitro model of the arterial network described by [Alastruey et al. 2011, J. Biomech. bf 44], for which experimental flow and pressure measurements are available at few selected locations. In order to mimic clinically relevant situations, we focus on the estimation of terminal resistances and arterial wall parameters related to vessel mechanics (Young's modulus and thickness) using few experimental observations (at most a single pressure or flow measurement per vessel). In all cases, we first perform a theoretical identifiability analysis based on the generalized sensitivity function, comparing then the results obtained with the ROUKF, using either synthetic or experimental data, to results obtained using reference parameters and to available measurements.

  • K. Disser, M. Liero, J. Zinsl, On the evolutionary Gamma-convergence of gradient systems modeling slow and fast chemical reactions, Preprint no. 2227, WIAS, Berlin, 2016.
    Abstract, PDF (489 kByte)
    We investigate the limit passage for a system of ordinary differential equations modeling slow and fast chemical reaction of mass-action type, where the rates of fast reactions tend to infinity. We give an elementary proof of convergence to a reduced dynamical system acting in the slow reaction directions on the manifold of fast reaction equilibria. Then we study the entropic gradient structure of these systems and prove an E-convergence result via Γ-convergence of the primary and dual dissipation potentials, which shows that this structure carries over to the fast reaction limit. We recover the limit dynamics as a gradient flow of the entropy with respect to a pseudo-metric.

  • W. Dreyer, C. Guhlke, M. Landstorfer, R. Müller, New insights on the interfacial tension of electrochemical interfaces and the Lippmann equation, Preprint no. 2201, WIAS, Berlin, 2015.
    Abstract, PDF (4623 kByte)
    The Lippmann equation is considered as universal relationship between interfacial tension, double layer charge, and cell potential. Based on the framework of continuum thermo-electrodynamics we provide some crucial new insights to this relation. In a previous work we have derived a general thermodynamic consistent model for electrochemical interfaces, which showed a remarkable agreement to single crystal experimental data. Here we apply the model to a curved liquid metal electrode. If the electrode radius is large compared to the Debye length, we apply asymptotic analysis methods and obtain the Lippmann equation. We give precise definitions of the involved quantities and show that the interfacial tension of the Lippmann equation is composed of the surface tension of our general model, and contributions arising from the adjacent space charge layers. This finding is confirmed by a comparison of our model to experimental data of several mercury-electrolyte interfaces. We obtain qualitative and quantitative agreement in the 2V potential range for various salt concentrations. We also discuss the validity of our asymptotic model when the electrode curvature radius is comparable to the Debye length.

  • M. Liero, S. Reichelt, Homogenization of Cahn--Hilliard-type equations via evolutionary Gamma-convergence, Preprint no. 2114, WIAS, Berlin, 2015.
    Abstract, PDF (455 kByte)
    In this paper we discuss two approaches to evolutionary Γ-convergence of gradient systems in Hilbert spaces. The formulation of the gradient system is based on two functionals, namely the energy functional and the dissipation potential, which allows us to employ Γ-convergence methods. In the first approach we consider families of uniformly convex energy functionals such that the limit passage of the time-dependent problems can be based on the theory of evolutionary variational inequalities as developed by Daneri and Savaré 2010. The second approach uses the equivalent formulation of the gradient system via the energy-dissipation principle and follows the ideas of Sandier and Serfaty 2004. We apply both approaches to rigorously derive homogenization limits for Cahn-Hilliard-type equations. Using the method of weak and strong two-scale convergence via periodic unfolding, we show that the energy and dissipation functionals Γ-converge. In conclusion, we will give specific examples for the applicability of each of the two approaches.

  Talks, Poster

  • K. Disser, E-convergence to the quasi-steady-state approximation in systems of chemical reactions, ERC Workshop on Modeling Materials and Fluids using Variational Methods, February 22 - 26, 2016, WIAS Berlin, Berlin, February 25, 2016.

  • S. Reichelt, Competing patterns in anti-symmetrically coupled Swift--Hohenberg equations, International Conference on Control of Complex Systems and Networks, SFB 910 ``Control of Self-Organizing Nonlinear Systems: Theoretical Methods and Concepts of Application'', Heringsdorf/Usedom, September 4 - 8, 2016.

  • M. Heida, Homogenization of the random conductance model, 7th European Congress of Mathematics (ECM), session ``Probability, Statistics and Financial Mathematics'', July 18 - 22, 2016, Technische Universität Berlin, Berlin, July 20, 2016.

  • M. Heida, Homogenization of the random conductance model, Workshop of the GAMM Activity Group ``Analysis of Partial Differential Equations'', September 26 - 28, 2016, Technische Universität Dortmund, Fachbereich Mathematik, Dortmund, September 26, 2016.

  • M. Heida, Stochastic homogenization of 1-homogeneous functionals, 7th European Congress of Mathematics (7ECM), Minisymposium 29 ``Nonsmooth PDEs in the Modeling Damage, Delamination, and Fracture'', July 18 - 22, 2016, Technische Universität Berlin, July 22, 2016.

  • M. Heida, Stochastic homogenization of rate-independent systems, Berlin Dresden Prague Würzburg Workshop ``Homogenization and Related Topics'', Technische Universität Dresden, Fachbereich Mathematik, June 22, 2016.

  • M. Heida, Stochastic homogenization of rate-independent systems, Joint Annual Meeting of DMV and GAMM, Session ``Multiscales and Homogenization'', March 7 - 11, 2016, Technische Universität Braunschweig, Braunschweig, March 10, 2016.

  • R. Müller, W. Dreyer, J. Fuhrmann, C. Guhlke, New insights into Butler--Volmer kinetics from thermodynamic modeling, The 67th Annual Meeting of the International Society of Electrochemistry, Den Haag, Netherlands, August 21 - 26, 2016.

  • J. Fuhrmann, Ch. Merdon, A thermodynamically consistent numerical approach to Nernst--Planck--Poisson systems with volume constraints, The 67th Annual Meeting of the International Society of Electrochemistry, Den Haag, Netherlands, August 21 - 26, 2016.

  • J. Fuhrmann, W. Dreyer, C. Guhlke, M. Landstorfer, R. Müller, A. Linke, Ch. Merdon, Modeling and numerics for electrochemical systems, Micro Battery and Capacitive Energy Harvesting Materials -- Results of the MatFlexEnd Project, Universität Wien, Austria, September 19, 2016.

  • C. Guhlke, J. Fuhrmann, W. Dreyer, R. Müller, M. Landstorfer, Modeling of batteries, Batterieforum Deutschland 2016, Berlin, April 6 - 8, 2016.

  • A. Caiazzo, Multiscale modeling of weakly compressible elastic materials in harmonic regime, Université de Franche-Comté, Laboratoire de Mathématiques de Besançon, France, July 1, 2014.

  • S. Neukamm, Optimal decay estimate on the semigroup associated with a random walk among random conductances, Dirichlet Forms and Applications, German-Japanese Meeting on Stochastic Analysis, September 9 - 13, 2013, Universität Leipzig, Mathematisches Institut, September 9, 2013.

  • H. Abels, J. Daube, Ch. Kraus, D. Kröner, Sharp interface limit for the Navier--Stokes--Korteweg model, DIMO2013 -- Diffuse Interface Models, Levico Terme, Italy, September 10 - 13, 2013.

  • CH. Kraus, Damage and phase separation processes: Modeling and analysis of nonlinear PDE systems, DIMO2013 -- Diffuse Interface Models, September 10 - 13, 2013, Levico Terme, Italy, September 11, 2013.

  • CH. Kraus, Modeling and analysis of a nonlinear PDE system for phase separation and damage, Università di Pavia, Dipartimento di Matematica, Italy, January 22, 2013.

  • CH. Kraus, Sharp interface limit of a diffuse interface model of Navier--Stokes--Allen--Cahn type for mixtures, Workshop ``Hyperbolic Techniques for Phase Dynamics'', June 10 - 14, 2013, Mathematisches Forschungsinstitut Oberwolfach, June 11, 2013.

  • CH. Kraus, A nonlinear PDE system for phase separation and damage, Universität Freiburg, Abteilung Angewandte Mathematik, November 13, 2012.

  • CH. Kraus, Cahn--Larché systems coupled with damage, Università degli Studi di Milano, Dipartimento di Matematica, Italy, November 28, 2012.

  • CH. Kraus, Phase field systems for phase separation and damage processes, 12th International Conference on Free Boundary Problems: Theory and Applications, June 11 - 15, 2012, Frauenchiemsee, June 12, 2012.

  • CH. Kraus, Phasenfeldsysteme für Entmischungs- und Schädigungsprozesse, Mathematisches Kolloquium, Universität Stuttgart, Fachbereich Mathematik, May 15, 2012.

  • CH. Kraus, Diffuse interface systems for phase separation and damage, Seminar on Partial Differential Equations, Institute of Mathematics, Academy of Sciences of the Czech Republic, Prague, May 3, 2011.

  • CH. Kraus, Phase separation systems coupled with elasticity and damage, ICIAM 2011, July 18 - 22, 2011, Vancouver, Canada, July 18, 2011.

  • W. Dreyer, On a paradox within the phase field modeling of storage systems and its resolution, 8th AIMS International Conference on Dynamical Systems, Differential Equations and Applications, May 25 - 28, 2010, Technische Universität Dresden, May 26, 2010.

  • CH. Kraus, An inhomogeneous, anisotropic and elastically modified Gibbs-Thomson law as singular limit of a diffuse interface model, 81st Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM), March 22 - 26, 2010, Karlsruhe, March 23, 2010.

  • CH. Kraus, Inhomogeneous and anisotropic phase-field quantities in the sharp interface limit, 6th Singular Days 2010, April 29 - May 1, 2010, WIAS, Berlin, April 30, 2010.

  • 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.

  • W. Dreyer, On a paradox within the phase field modeling of storage systems and its resolution, PF09 -- 2nd Symposium on Phase-Field Modelling in Materials Science, August 30 - September 2, 2009, Universität Aachen, Kerkrade, Netherlands, August 31, 2009.

  • W. Dreyer, Phase field models and their sharp limits in the context of hydrogen storage and lithium-ion batteries, 1textsuperscriptst International Conference on Material Modelling (1textsuperscriptst ICMM), September 15 - 18, 2009, Technische Universität Dortmund, September 16, 2009.

  • CH. Kraus, A phase-field model with anisotropic surface tension in the sharp interface limit, Second GAMM-Seminar on Multiscale Material Modelling, July 10 - 12, 2008, Universität Stuttgart, Institut für Mechanik (Bauwesen), July 12, 2008.

  • CH. Kraus, Ein Phasenfeldmodell vom Cahn-Hilliard-Typ im singulären Grenzwert, Oberseminar Analysis, Universität Regensburg, Fakultät für Mathematik, April 25, 2008.

  • CH. Kraus, Phase field models and corresponding Gibbs--Thomson laws. Part II, SIMTECH Seminar Multiscale Modelling in Fluid Mechanics, Universität Stuttgart, Institut für Angewandte Analysis und Numerische Simulation, November 5, 2008.

  • CH. Kraus, On jump conditions at phase interfaces, Oberseminar über Angewandte Mathematik, December 10 - 15, 2007, Universität Freiburg, Abteilung für Angewandte Mathematik, December 11, 2007.

  • CH. Kraus, Equilibrium conditions for liquid-vapor system in the sharp interface limit, Seminar Thermodynamische Modellierung und Analyse von Phasenübergängen, WIAS, Berlin, July 18, 2006.

  • CH. Kraus, Equilibria conditions in the sharp interface limit of the van der Waals-Cahn-Hilliard phase model, Recent Advances in Free Boundary Problems and Related Topics (FBP2006), September 14 - 16, 2006, Levico, Italy, September 14, 2006.

  • CH. Kraus, The sharp interface limit of the van der Waals--Cahn--Hilliard model, Polish-German Workshop ``Modeling Structure Formation'', Interdisciplinary Centre for Mathematical and Computational Modelling, University of Warsaw, Poland, September 8, 2006.

  • CH. Kraus, On the sharp limit of the Van der Waals-Cahn-Hilliard model, WIAS Workshop ``Dynamic of Phase Transitions'', November 30 - December 3, 2005, Berlin, December 2, 2005.

  • CH. Kraus, On the sharp limit of the Van der Waals-Cahn-Hilliard model, Workshop ``Micro-Macro Modeling and Simulation of Liquid-Vapor Flows'', November 16 - 18, 2005, Universität Freiburg, Mathematisches Institut, Kirchzarten, November 17, 2005.

  • CH. Kraus, Maximale Konvergenz in höheren Dimensionen, Seminar Thermodynamische Modellierung und Analyse von Phasenübergängen, WIAS, Berlin, May 24, 2005.