Some of the current members of the research group "Thermodynamic Modeling and Analysis of Phase Transitions" were members of a former group or of RG 1 respectively. Therefore, the corresponding publications can found on the web pages of these groups:

  • former Young Scientists' Group "Modeling of Damage Processes"
  • former Leibniz Group "Mathematical Models for Lithium-Ion Batteries"
  • Research group 1 "Partial Differential Equations"
    • Monographs

      • S. Jachalski, D. Peschka, S. Bommer, R. Seemann, B. Wagner, Chapter 18: Structure Formation in Thin Liquid--Liquid Films, in: Transport Processes at Fluidic Interfaces, D. Bothe, A. Reusken, eds., Advances in Mathematical Fluid Mechanics, Birkhäuser, Springer International Publishing AG, Cham, 2017, pp. 531--574, (Chapter Published), DOI 10.1007/978-3-319-56602-3 .
        We revisit the problem of a liquid polymer that dewets from another liquid polymer substrate with the focus on the direct comparison of results from mathematical modeling, rigorous analysis, numerical simulation and experimental investigations of rupture, dewetting dynamics and equilibrium patterns of a thin liquid-liquid system. The experimental system uses as a model system a thin polystyrene (PS) / polymethylmethacrylate (PMMA) bilayer of a few hundred nm. The polymer systems allow for in situ observation of the dewetting process by atomic force microscopy (AFM) and for a precise ex situ imaging of the liquid--liquid interface. In the present study, the molecular chain length of the used polymers is chosen such that the polymers can be considered as Newtonian liquids. However, by increasing the chain length, the rheological properties of the polymers can be also tuned to a viscoelastic flow behavior. The experimental results are compared with the predictions based on the thin film models. The system parameters like contact angle and surface tensions are determined from the experiments and used for a quantitative comparison. We obtain excellent agreement for transient drop shapes on their way towards equilibrium, as well as dewetting rim profiles and dewetting dynamics.

      Articles in Refereed Journals

      • W. Dreyer, P. Friz, P. Gajewski, C. Guhlke, M. Maurelli, Stochastic many-particle model for LFP electrodes, Continuum Mechanics and Thermodynamics, 30 (2018), pp. 593--628, DOI 10.1007/s00161-018-0629-7 .
        In the framework of non-equilibrium thermodynamics we derive a new model for porous electrodes. The model is applied to LiFePO4 (LFP) electrodes consisting of many LFP particles of nanometer size. The phase transition from a lithium-poor to a lithium-rich phase within LFP electrodes is controlled by surface fluctuations leading to a system of stochastic differential equations. The model is capable to derive an explicit relation between battery voltage and current that is controlled by thermodynamic state variables. This voltage-current relation reveals that in thin LFP electrodes lithium intercalation from the particle surfaces into the LFP particles is the principal rate limiting process. There are only two constant kinetic parameters in the model describing the intercalation rate and the fluctuation strength, respectively. The model correctly predicts several features of LFP electrodes, viz. the phase transition, the observed voltage plateaus, hysteresis and the rate limiting capacity. Moreover we study the impact of both the particle size distribution and the active surface area on the voltagecharge characteristics of the electrode. Finally we carefully discuss the phase transition for varying charging/discharging rates.

      • M. Landstorfer, On the dissociation degree of ionic solutions considering solvation effects, Electrochemistry Communications, 92 (2018), pp. 56--59, DOI 10.20347/WIAS.PREPRINT.2443 .
        In this work the impact of solvation effects on the dissociation degree of strong electrolytes and salts is discussed. The investigation is based on a thermodynamic model which is capable to predict qualitatively and quantitatively the double layer capacity of various electrolytes. A remarkable relationship between capacity maxima, partial molar volume of ions in solution, and solvation numbers, provides an experimental access to determine the number of solvent molecules bound to a specific ion in solution. This shows that the Stern layer is actually a saturated solution of 1 mol L-1 solvated ions, and we point out some fundamental similarities of this state to a saturated bulk solution. Our finding challenges the assumption of complete dissociation, even for moderate electrolyte concentrations, whereby we introduce an undissociated ion-pair in solution. We re-derive the equilibrium conditions for a two-step dissociation reaction, including solvation effects, which leads to a new relation to determine the dissociation degree. A comparison to Ostwald's dilution law clearly shows the shortcomings when solvation effects are neglected and we emphasize that complete dissociation is questionable beyond 0.5 mol L-1 for aqueous, mono-valent electrolytes.

      • T. Ahnert, A. Münch, B. Niethammer, B. Wagner, Stability of concentrated suspensions under Couette and Poiseuille flow, Journal of Engineering Mathematics, (2018), published online on 21.02.2018, DOI 10.1007/s10665-018-9954-x .
        The stability of two-dimensional Poiseuille flow and plane Couette flow for concentrated suspensions is investigated. Linear stability analysis of the two-phase flow model for both flow geometries shows the existence of a convectively driven instability with increasing growth rates of the unstable modes as the particle volume fraction of the suspension increases. In addition it is shown that there exists a bound for the particle phase viscosity below which the two-phase flow model may become ill-posed as the particle phase approaches its maximum packing fraction. The case of two-dimensional Poiseuille flow gives rise to base state solutions that exhibit a jammed and unyielded region, due to shear-induced migration, as the maximum packing fraction is approached. The stability characteristics of the resulting Bingham-type flow is investigated and connections to the stability problem for the related classical Bingham-flow problem are discussed.

      • T. Ahnert, A. Münch, B. Wagner, Models for the two-phase flow of concentrated suspensions, European Journal of Applied Mathematics, (2018), published online on 04.06.2018, DOI 10.1017/S095679251800030X .
        A new two-phase model is derived that make use of a constitutive law combining non-Brownian suspension with granular rheology, that was recently proposed by Boyer et al. [PRL, 107(18),188301 (2011)]. It is shown that for the simple channel flow geometry, the stress model naturally exhibits a Bingham type flow property with an unyielded finite-size zone in the center of the channel. As the volume fraction of the solid phase is increased, the various transitions in the flow fields are discussed using phase space methods for a boundary value problem, that is derived from the full model. The predictions of this analysis is then compared to the direct finite-element numerical solutions of the full model.

      • W. Dreyer, C. Guhlke, R. Müller, Bulk-surface electro-thermodynamics and applications to electrochemistry, Entropy. An International and Interdisciplinary Journal of Entropy and Information Studies, 20 (2018), 939, DOI 10.3390/e20120939 .
        We propose a modeling framework for magnetizable, polarizable, elastic, viscous, heat conducting, reactive mixtures in contact with interfaces. To this end we first introduce bulk and surface balance equations that contain several constitutive quantities. For further modeling the constitutive quantities, we formulate constitutive principles. They are based on an axiomatic introduction of the entropy principle and the postulation of Galilean symmetry. We apply the proposed formalism to derive constitutive relations in a rather abstract setting. For illustration of the developed procedure, we state an explicit isothermal material model for liquid electrolyte metal electrode interfaces in terms of free energy densities in the bulk and on the surface. Finally we give a survey of recent advancements in the understanding of electrochemical interfaces that were based on this model.

      • 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.

      • W. Dreyer, C. Guhlke, Sharp limit of the viscous Cahn--Hilliard equation and thermodynamic consistency, Continuum Mechanics and Thermodynamics, 29 (2017), pp. 913--934.
        Diffuse and sharp interface models represent two alternatives to describe phase transitions with an interface between two coexisting phases. The two model classes can be independently formulated. Thus there arises the problem whether the sharp limit of the diffuse model fits into the setting of a corresponding sharp interface model. We call a diffuse model admissible if its sharp limit produces interfacial jump conditions that are consistent with the balance equations and the 2nd law of thermodynamics for sharp interfaces. We use special cases of the viscous Cahn-Hilliard equation to show that there are admissible as well as non-admissible diffuse interface models.

      • M. Eigel, R. Müller, A posteriori error control for stationary coupled bulk-surface equations, IMA Journal of Numerical Analysis, 38 (2018), pp. 271--298 (published online on 09.03.2017), DOI 10.1093/imanum/drw080 .
        We consider a system of two coupled elliptic equations, one defined on a bulk domain and the other one on the boundary surface. Problems of this kind are relevant for applications in engineering, chemistry and in biology like e.g. biological signal transduction. For the a posteriori error control of the coupled system, a residual error estimator is derived which takes into account the approximation errors due to the finite element discretisation in space as well as the polyhedral approximation of the surface. An adaptive refinement algorithm controls the overall error. Numerical experiments illustrate the performance of the a posteriori error estimator and the adaptive algorithm with several benchmark examples.

      • M. Landstorfer, Boundary conditions for electrochemical interfaces, Journal of The Electrochemical Society, 164 (2017), pp. 3671--3685.
        Consistent boundary conditions for electrochemical interfaces, which cover double layer charging, pseudo-capacitive effects and transfer reactions, are of high demand in electrochemistry and adjacent disciplines. Mathematical modeling and optimization of electrochemical systems is a strongly emerging approach to reduce cost and increase efficiency of super-capacitors, batteries, fuel cells, and electrocatalysis. However, many mathematical models which are used to describe such systems lack a real predictive value. Origin of this shortcoming is the usage of oversimplified boundary conditions. In this work we derive the boundary conditions for some general electrode-electrolyte interface based on non-equilibrium thermodynamics for volumes and surfaces. The resulting equations are widely applicable and cover also tangential transport. The general framework is then applied to a specific material model which allows the deduction of a current-voltage relation and thus a comparison to experimental data. Some simplified 1D examples show the range of applicability of the new approach.

      • E. Meca Álvarez, V. Shenoy , J. Lowengrub, H2-dependent attachment kinetics and shape evolution in CVD graphene growth, 2D Materials, 4 (2017), pp. 031010/1--031010/5, DOI 10.1088/2053-1583/aa74f1 .

      • 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.

      • CH. Heinemann, Ch. Kraus, E. Rocca, R. Rossi, A temperature-dependent phase-field model for phase separation and damage, Archive for Rational Mechanics and Analysis, 225 (2017), pp. 177--247.
        In this paper we study a model for phase separation and damage in thermoviscoelastic materials. The main novelty of the paper consists in the fact that, in contrast with previous works in the literature (cf., e.g., [C. Heinemann, C. Kraus: Existence results of weak solutions for Cahn-Hilliard systems coupled with elasticity and damage. Adv. Math. Sci. Appl. 21 (2011), 321--359] and [C. Heinemann, C. Kraus: Existence results for diffuse interface models describing phase separation and damage. European J. Appl. Math. 24 (2013), 179--211]), we encompass in the model thermal processes, nonlinearly coupled with the damage, concentration and displacement evolutions. More in particular, we prove the existence of "entropic weak solutions", resorting to a solvability concept first introduced in [E. Feireisl: Mathematical theory of compressible, viscous, and heat conducting fluids. Comput. Math. Appl. 53 (2007), 461--490] in the framework of Fourier-Navier-Stokes systems and then recently employed in [E. Feireisl, H. Petzeltová, E. Rocca: Existence of solutions to a phase transition model with microscopic movements. Math. Methods Appl. Sci. 32 (2009), 1345--1369], [E. Rocca, R. Rossi: "Entropic" solutions to a thermodynamically consistent PDE system for phase transitions and damage. SIAM J. Math. Anal., 47 (2015), 2519--2586] for the study of PDE systems for phase transition and damage. Our global-in-time existence result is obtained by passing to the limit in a carefully devised time-discretization scheme.

      • A. Münch, B. Wagner, L.P. Cook, R.R. Braun, Apparent slip for an upper convected Maxwell fluid, SIAM Journal on Applied Mathematics, 77 (2017), pp. 537--564, DOI 10.1137/16M1056869 .
        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. Roggensack, Ch. Kraus, Existence of weak solutions for the Cahn--Hilliard reaction model including elastic effects and damage, Journal of Partial Differential Equations, 30 (2017), pp. 111-145, DOI 10.4208/jpde.v30.n2.2 .
        In this paper, we introduce and study analytically a vectorial Cahn-Hilliard reaction model coupled with rate-dependent damage processes. The recently proposed Cahn-Hilliard reaction model can e.g. be used to describe the behavior of electrodes of lithium-ion batteries as it includes both the intercalation reactions at the surfaces and the separation into different phases. The coupling with the damage process allows considering simultaneously the evolution of a damage field, a second important physical effect occurring during the charging or discharging of lithium-ion batteries. Mathematically, this is realized by a Cahn-Larché system with a non-linear Newton boundary condition for the chemical potential and a doubly non-linear differential inclusion for the damage evolution. We show that this system possesses an underlying generalized gradient structure which incorporates the non-linear Newton boundary condition. Using this gradient structure and techniques from the field of convex analysis we are able to prove constructively the existence of weak solutions of the coupled PDE system.

      • CH. Kraus, M. Radszuweit, Modeling and simulation of non-isothermal rate-dependent damage processes in inhomogeneous materials using the phase-field approach, Computational Mechanics, 60 (2017), pp. 163--179, DOI 10.1007/s00466-017-1393-4 .
        We present a continuum model that incorporates rate-dependent damage and fracture, a material order parameter field and temperature. Different material characteristics throughout the medium yield a strong inhomogeneity and affect the way fracture propagates. The phasefield approach is employed to describe degradation. For the material order parameter we assume a Cahn Larché-type dynamics, which makes the model in particular applicable to binary alloys. We give thermodynamically consistent evolution equations resulting from a unified variational approach. Diverse coupling mechanisms can be covered within the model, such as heat dissipation during fracture, thermal-expansion-in- duced failure and elastic-inhomogeneity effects. We furthermore present an adaptive Finite Element code in two space dimensions, that is capable of solving such a highly nonlinear and non-convex system of partial differential equations. With the help of this tool we conduct numerical experiments of different complexity in order to investigate the possibilities and limitations of the presented model. A main feature of our model is that we can describe the process of micro-crack nucleation in regions of partial damage to form macro-cracks in a unifying approach.

      • W. Dreyer, C. Guhlke, M. Landstorfer, R. Müller, New insights on the interfacial tension of electrochemical interfaces and the Lippmann equation, European Journal of Applied Mathematics, 29 (2018), pp. 708--753, DOI 10.1017/S0956792517000341 .
        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.

      • E. Meca Álvarez, A. Münch, B. Wagner, Sharp-interface formation during lithium intercalation into silicon, European Journal of Applied Mathematics, 29 (2018), pp. 118--145, DOI 10.1017/S0956792517000067 .
        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.

      Contributions to Collected Editions

      • J. Fuhrmann, C. Guhlke, A finite volume scheme for Nernst--Planck--Poisson systems with Ion size and solvation effects, in: Finite Volumes for Complex Applications VIII -- Hyperbolic, Elliptic and Parabolic Problems, FVCA 8, Lille, France, June 2017, C. Cancès, P. Omnes, eds., 200 of Springer Proceedings in Mathematics & Statistics, Springer International Publishing AG, Cham, 2017, pp. 497--505, DOI 10.1007/978-3-319-57394-6_52 .

      Preprints, Reports, Technical Reports

      • P. Nestler, N. Schlömer, O. Klein, J. Sprekels, F. Tröltzsch, Optimal control of semiconductor melts by traveling magnetic fields, Preprint no. 2549, WIAS, Berlin, 2018, DOI 10.20347/WIAS.PREPRINT.2549 .
        Abstract, PDF (2938 kByte)
        In this paper, the optimal control of traveling magnetic fields in a process of crystal growth from the melt of semiconductor materials is considered. As controls, the phase shifts of the voltage in the coils of a heater-magnet module are employed to generate Lorentz forces for stirring the crystal melt in an optimal way. By the use of a new industrial heater-magnet module, the Lorentz forces have a stronger impact on the melt than in earlier technologies. It is known from experiments that during the growth process temperature oscillations with respect to time occur in the neighborhood of the solid-liquid interface. These oscillations may strongly influence the quality of the growing single crystal. As it seems to be impossible to suppress them completely, the main goal of optimization has to be less ambitious, namely, one tries to achieve oscillations that have a small amplitude and a frequency which is sufficiently high such that the solid-liquid interface does not have enough time to react to the oscillations. In our approach, we control the oscillations at a finite number of selected points in the neighborhood of the solidification front. The system dynamics is modeled by a coupled system of partial differential equations that account for instationary heat condution, turbulent melt flow, and magnetic field. We report on numerical methods for solving this system and for the optimization of the whole process. Different objective functionals are tested to reach the goal of optimization.

      • 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

      • W. Dreyer, C. Guhlke, R. Müller, The impact of solvation and dissociation on the transport parameters of liquid electrolytes: Continuum modeling and numerical study, Preprint no. 2539, WIAS, Berlin, 2018, DOI 10.20347/WIAS.PREPRINT.2539 .
        Abstract, PDF (1472 kByte)
        Electro-thermodynamics provides a consistent framework to derive continuum models for electrochemical systems. For the application to a specific experimental system, the general model must be equipped with two additional ingredients: a free energy model to calculate the chemical potentials and a kinetic model for the kinetic coefficients. Suitable free energy models for liquid electrolytes incorporating ion-solvent interaction, finite ion sizes and solvation already exist and have been validated against experimental measurements. In this work, we focus on the modeling of the mobility coefficients based on Maxwell--Stefan setting and incorporate them into the general electro-thermodynamic framework. Moreover, we discuss the impact of model parameter on conductivity, transference numbers and salt diffusion coefficient. In particular, the focus is set on the solvation of ions and incomplete dissociation of a non-dilute electrolyte.

      • 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.

      • M. Landstorfer, The dielectric constant of liquid electrolytes obtained from periodic homogenization theory, Preprint no. 2531, WIAS, Berlin, 2018, DOI 10.20347/WIAS.PREPRINT.2531 .
        Abstract, PDF (6771 kByte)
        The dielectric constant of an electrolytic solution is known to decrease with increasing salt concentration. This effect, frequently called dielectric decrement, is experimentally found for many salts and solvents and shows an almost linear decrease up to a certain salt concentration. However, the actual origin of this concentration dependence is yet unclear, and many different theoretical approaches investigate this effect. Here I present an investigation based on microscopic Maxwell equations and periodic homogenization theory. The microscopic perception of anions and cations forming a pseudo lattice in the liquid solution is exploited by multi-scale asymptotic expansions, where the inverse Avogadro number arises as small scaling parameter. This leads to a homogenized Poisson equation on the continuum scale with an effective or homogenized dielectric constant that accounts for microscopic field effects in the pseudo lattice. Incomplete dissociation is further considered at higher salt concentrations due to solvation effects. The numerically computed homogenized dielectric constant is then compared to experimental data of NaCl and shows a remarkable qualitative and quantitative agreement in the concentration range of (0 - 5)mol L.

      • J. Fuhrmann, C. Guhlke, B. Matejczyk, R. Müller, Transport of solvated ions in nanopores: Asymptotic models and numerical study, Preprint no. 2526, WIAS, Berlin, 2018, DOI 10.20347/WIAS.PREPRINT.2526 .
        Abstract, PDF (1233 kByte)
        Improved Poisson--Nernst--Planck systems taking into account finite ion size and solvation effects provide a more accurate model of electric double layers compared to the classical setting. We introduce and discuss several variants of such improved models. %Based on spatially fully resolved numerical models We study the effect of improved modeling in large aspect ratio nanopores. Moreover, we derive approximate asymptotic models for the improved Poisson--Nernst--Planck systems which can be reduced to one-dimensional systems. In a numerical study, we compare simulation results obtained from solution of the asymptotic 1D-models with those obtained by discretization of the full resolution models.

      • J. Fuhrmann, C. Guhlke, A. Linke, Ch. Merdon, R. Müller, Models and numerical methods for electrolyte flows, Preprint no. 2525, WIAS, Berlin, 2018, DOI 10.20347/WIAS.PREPRINT.2525 .
        Abstract, PDF (1807 kByte)
        The most common mathematical models for electrolyte flows are based on the dilute solution assumption, leading to a coupled system of the Nernst--Planck--Poisson drift-diffusion equations for ion transport and the Stokes resp. Navier--Stokes equations for fluid flow. This contribution discusses historical and recent model developments beyond the dilute solution assumption and focuses on the effects of finite ion sizes and solvation. A novel numerical solution approach is presented and verified here which aims at preserving on the discrete level consistency with basic thermodynamic principles and structural properties like independence of flow velocities from gradient contributions to external forces.

      • P.-É. Druet, Global--in--time solvability of thermodynamically motivated parabolic systems, Preprint no. 2455, WIAS, Berlin, 2017, DOI 10.20347/WIAS.PREPRINT.2455 .
        Abstract, PDF (283 kByte)
        In this paper, doubly non linear parabolic systems in divergence form are investigated form the point of view of their global--in--time weak solvability. The non--linearity under the time derivative is given by the gradient of a strictly convex, globally Lipschitz continuous potential, multiplied by a position--dependent weight. This weight admits singular values. The flux under the spatial divergence is also of monotone gradient type, but it is not restricted to polynomial growth. It is assumed that the elliptic operator generates some equi--coercivity on the spatial derivatives of the unknowns. The paper introduces some original techniques to deal with the case of nonlinear purely Neumann boundary conditions. In this respect, it generalises or complements the results by Alt and Luckhaus (1983) and Alt (2012). A field of application of the theory are the multi species diffusion systems driven by entropy.

      • P.-É. Druet, Local well-posedness for thermodynamically motivated quasilinear parabolic systems in divergence form, Preprint no. 2454, WIAS, Berlin, 2017, DOI 10.20347/WIAS.PREPRINT.2454 .
        Abstract, PDF (367 kByte)
        We show that fully quasilinear parabolic systems are locally well posed in the Hilbert space scala if the coefficients of the differential operator are smooth enough and the spatial domain is sufficiently regular. In the context of diffusion systems driven by entropy, the uniform parabolicity follows from the second law of thermodynamics.

      • G. Kitavtsev, A. Münch, B. Wagner, Thin film models for an active gel, Preprint no. 2451, WIAS, Berlin, 2017, DOI 10.20347/WIAS.PREPRINT.2451 .
        Abstract, PDF (321 kByte)
        In this study we present a free-boundary problem for an active liquid crystal based on the Beris-Edwards theory that uses a tensorial order parameter and includes active contributions to the stress tensor to analyse the rich defect structure observed in applications such as the Adenosinetriphosphate (ATP) driven motion of a thin film of an actin filament network. The small aspect ratio of the film geometry allows for an asymptotic approximation of the free-boundary problem in the limit of weak elasticity of the network and strong active terms. The new thin film model captures the defect dynamics in the bulk as well as wall defects and thus presents a significant extension of previous models based on the Leslie-Erickson-Parodi theory. Analytic expressions are derived that reveal the interplay of anchoring conditions, film thickness and active terms and their control of transitions of flow structure.

      • W. Dreyer, P.-É. Druet, P. Gajewski, C. Guhlke, Analysis of improved Nernst--Planck--Poisson models of compressible isothermal electrolytes. Part III: Compactness and convergence, Preprint no. 2397, WIAS, Berlin, 2017, DOI 10.20347/WIAS.PREPRINT.2397 .
        Abstract, PDF (327 kByte)
        We consider an improved Nernst--Planck--Poisson model first proposed by Dreyer et al. in 2013 for compressible isothermal electrolytes in non equilibrium. The model takes into account the elastic deformation of the medium that induces an inherent coupling of mass and momentum transport. The model consists of convection--diffusion--reaction equations for the constituents of the mixture, of the Navier-Stokes equation for the barycentric velocity, and of the Poisson equation for the electrical potential. Due to the principle of mass conservation, cross--diffusion phenomena must occur and the mobility matrix (Onsager matrix) has a kernel. In this paper, which continues the investigations of [DDGG17a, DDGG17b], we prove the compactness of the solution vector, and existence and convergence for the approximation schemes. We point at simple structural PDE arguments as an adequate substitute to the Aubin--Lions compactness Lemma and its generalisations: These familiar techniques attain their limit in the context of our model in which the relationship between time derivatives (transport) and diffusion gradients is highly non linear.

      • W. Dreyer, P.-É. Druet, P. Gajewski, C. Guhlke, Analysis of improved Nernst--Planck--Poisson models of compressible isothermal electrolytes. Part II: Approximation and a priori estimates, Preprint no. 2396, WIAS, Berlin, 2017, DOI 10.20347/WIAS.PREPRINT.2396 .
        Abstract, PDF (355 kByte)
        We consider an improved Nernst--Planck--Poisson model first proposed by Dreyer et al. in 2013 for compressible isothermal electrolytes in non equilibrium. The model takes into account the elastic deformation of the medium that induces an inherent coupling of mass and momentum transport. The model consists of convection--diffusion--reaction equations for the constituents of the mixture, of the Navier-Stokes equation for the barycentric velocity, and of the Poisson equation for the electrical potential. Due to the principle of mass conservation, cross--diffusion phenomena must occur and the mobility matrix (Onsager matrix) has a kernel. In this paper, which continues the investigation of [DDGG17a], we derive for thermodynamically consistent approximation schemes the natural uniform estimates associated with the dissipations. Our results essentially improve our former study [DDGG16], in particular the a priori estimates concerning the relative chemical potentials.

      • W. Dreyer, P.-É. Druet, P. Gajewski, C. Guhlke, Analysis of improved Nernst--Planck--Poisson models of compressible isothermal electrolytes. Part I: Derivation of the model and survey of the results, Preprint no. 2395, WIAS, Berlin, 2017, DOI 10.20347/WIAS.PREPRINT.2395 .
        Abstract, PDF (343 kByte)
        We consider an improved Nernst--Planck--Poisson model first proposed by Dreyer et al. in 2013 for compressible isothermal electrolytes in non equilibrium. The model takes into account the elastic deformation of the medium that induces an inherent coupling of mass and momentum transport. The model consists of convection--diffusion--reaction equations for the constituents of the mixture, of the Navier-Stokes equation for the barycentric velocity, and of the Poisson equation for the electrical potential. Due to the principle of mass conservation, cross--diffusion phenomena must occur and the mobility matrix (Onsager matrix) has a kernel. In this paper we establish the existence of a global--in--time weak solution for the full model, allowing for a general structure of the mobility tensor and for chemical reactions with highly non linear rates in the bulk and on the active boundary. We characterise the singular states of the system, showing that the chemical species can vanish only globally in space, and that this phenomenon must be concentrated in a compact set of measure zero in time. With respect to our former study [DDGG16], we also essentially improve the a priori estimates, in particular concerning the relative chemical potentials.

      • 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.

      Talks, Poster

      • P.-É. Druet, Analysis of mass transfer, Navier--Stokes equations for multicomponent fluids subject to a volume constraint, Berliner Oberseminar ``Nichtlineare partielle Differentialgleichungen'' (Langenbach-Seminar), WIAS, Berlin, November 28, 2018.

      • M. Landstorfer, Continuum thermodynamic modelling of electrolytes, BMBF Kickoff Meeting LuCaMag, Bonn, November 19, 2018.

      • M. Landstorfer, Homogenization methods for electrochemical systems, DFG Workshop Numerical Optimization of the PEM Fuel Cell Bipolar Plate, Universität Ulm, March 20, 2018.

      • M. Landstorfer, Modeling and simulation of porous battery electrodes with multi-scale homogenization techniques, ISE, Bologna, Italy, September 10, 2018.

      • M. Landstorfer, Modellbasierte Abschätzung der Lebensdauer von gealterten Li-Batterien für die 2nd-Life Anwendung als stationärer Stromspeicher, BMBF-Statusseminar zur Förderrichtlinie Mathematik, Bonn, November 26 - 27, 2018.

      • M. Landstorfer, Modellbasierte Abschätzung der Lebensdauer von gealterten Li-Batterien für die 2nd-Life Anwendung als stationärer Stromspeicher, BMBF-Statusseminar zur Förderrichtlinie Mathematik, Bonn, November 26, 2018.

      • M. Landstorfer, Modellierung und Upscaling, WIAS (AG 4), BMBF Workshop zu MaLLi2, November 12 - 13, 2018, Ellwangen, November 12, 2018.

      • M. Landstorfer, Modelling and simulation of porous battery electrodes with multi-scale homogenisation techniques, 6th European Conference on Computational Mechanics, 7th European Conference on Computational Fluid Dynamics (ECCM-ECFD 2018), June 11 - 15, 2018, Glasgow, UK, June 14, 2018.

      • M. Landstorfer, Modelling and simulation of porous battery electrodes with multi-scale homogenisation techniques, Solid state Electrochemistry Symposium, Hamburg, November 5, 2018.

      • M. Landstorfer, Modelling and simulation of porous electrodes with multi-scale homogenization technique, ModVal 2018, 15th Symposium on Modeling and Experimental Validation of Electrochemical Energy Devices, Aarau, Switzerland, April 12 - 13, 2018.

      • M. Landstorfer, Modelling battery electrodes with homogenization techniques, Kick-Off-Meeting zu BMBF-Projekt MALLi^2, Universität Ulm, March 21, 2018.

      • M. Landstorfer, Thermodynamic modeling of electrolytes and their boundary conditions to electrodes, AMaSiS 2018: Applied Mathematics and Simulation for Semiconductors, October 8 - 10, 2018, WIAS, Berlin, October 9, 2018.

      • R. Müller, J. Fuhrmann, C. Guhlke, B. Matejczyk , Dimension reduction of improved Nernst-Planck models for charged nanopores, Asymptotic Behavior of systems of PDE arising in physics and biology: theoretical and numerical points of view (ABPDE III), France, August 28 - 31, 2018.

      • R. Müller, Dynamics of electrochemical interfaces, Asymptotic Behavior of systems of PDE arising in physics and biology: theoretical and numerical points of view (ABPDE III), August 28 - 31, 2018, Universität Lille I, France, August 30, 2018.

      • R. Müller, Modeling and Simulation of electrolyte transport in nanopores, Center for Computational Engineering, RWTH Aachen, May 7, 2018.

      • 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.

      • B. Wagner, Modeling microstructures for light harvesting surfaces, ECMI 2018, 20th European Conference on Mathematics for Industry, June 18 - 22, 2018, Budapest, Hungary, June 21, 2018.

      • B. Wagner, Multiple scales in thin liquid films, UCLA Guest Lecture, University of California, Department of Mathematics, Los Angeles, USA, January 25, 2018.

      • B. Wagner, TPlenarvortrag SIAM Titel ergänzen, SIAM.

      • B. Wagner, Titel, Universität Münster, May 17, 2018.

      • B. Wagner, Titel, University Limerick, Ireland, June 11, 2018.

      • B. Wagner, Titel ergänzen, 89th Annual Meeting of the International Association of Applied Mathematics and Mechanics, March 19 - 23, 2018, Technische Universität München.

      • B. Wagner, Vortrag Minisymposium SIAM Titel ergänzen, SIAM.

      • B. Wagner, Vortragstitel ergänzen, Graz, Austria, September 10, 2018.

      • B. Wagner, Yield stress in concentrated suspensions, Mathematical Nanosystems Workshop, January 17 - 18, 2018, CNSI at UCLA, Los Angeles, USA, January 18, 2018.

      • O. Klein, On uncertainty quantification for models involving hysteresis operators, MURPHYS-HSFS-2018: Interdisciplinary Workshop on Multiple Scale Systems, Systems with Hysteresis and Trends in Dynamical Systems, May 28 - June 1, 2018, Centre de Recerca Matemàtica (CRM), Barcelona, Spain, May 31, 2018.

      • S. Bergmann, Phase-field modeling of Si thin-film growth via liquid-phase crystallization, Winter Workshop on Microstructure Characterization and Modeling for Solar Cells, February 12 - 16, 2017, Schliersee-Spitzingsee, February 13, 2017.

      • W. Dreyer, J. Fuhrmann, P. Gajewski, C. Guhlke, M. Landstorfer, M. Maurelli, R. Müller, Stochastic model for LiFePO4-electrodes, ModVal14 -- 14th Symposium on Fuel Cell and Battery Modeling and Experimental Validation, Karlsruhe, March 2 - 3, 2017.

      • M. Landstorfer, Theory, structure and experimental justification of the metal/electrolyte interface, Universität Münster, Institut für Analysis und Numerik, July 11, 2017.

      • R. Müller, A posteriori analysis for coupled bulk-surface problems, Oberseminar ``Angewandte Analysis und Numerische Simulation'', Universität Stuttgart, Institut für Angewandte Analysis und Numerische Simulation, June 1, 2017.

      • R. Müller, A posteriori error analysis for coupled bulk-surface diffusion, Matheon Workshop RMMM 8 - Berlin 2017, Reliable Methods of Mathematical Modeling, July 31 - August 3, 2017, Humboldt-Universität zu Berlin, July 31, 2017.

      • R. Müller, Consistent coupling of charge transport and fluid flow with application to nanopores, ACOMEN 2017 -- 7th International Conference on Advanced Computational Methods in Engineering, Minisymposium MS7 ``Electrokinetic and Electrochemical Flows for Batteries and Fuel Cells: Analysis, Simulation, Upscaling'', September 18 - 22, 2017, Ghent University, Belgium, September 21, 2017.

      • A. Roggensack, Damage processes in lithium-ion batteries, Workshop ``Recent Trends and Future Developments in Computational Science & Engineering, March 22 - 24, 2017, Plön, March 23, 2017.

      • B. Wagner, Mathematical opportunities and challenges in sustainable energies, SIAM Annual Meeting, July 10 - 14, 2017, Pittsburgh, USA, July 14, 2017.

      • B. Wagner, Rheologies of dense suspensions, Workshop ``Form and Deformation in Solid and Fluid Mechanics'', Isaac Newton Institute, Cambridge, UK, September 18 - 22, 2017.

      • B. Wagner, Yield stress for two-phase flow of concentrated suspensions, CIM-WIAS Workshop ``Topics in Applied Analysis and Optimisation'', December 6 - 8, 2017, International Center for Mathematics, University of Lisbon, Portugal, December 7, 2017.

      • C. Guhlke, Gefahr erkannt, Gefahr gebannt! - Mathematische Modelle machen Lithium-Ionen-Batterien sicherer, Symposium 25 Jahre Forschungsverbund Berlin e.V., Urania, Berlin, May 18, 2017.

      • C. Guhlke, Modelling of ion transport in electrolytes -- A thermodynamic approach, 2nd Dresden Battery Days, Dresden, September 18 - 20, 2017.

      • C. Guhlke, Vom Luftballon zur Lithium-Ionen-Batterie, Lange Nacht der Wissenschaften, Technische Universität Berlin, Haus der Mathematik, June 24, 2017.

      • O. Klein, Uncertainty quantification for models involving hysteresis operators, Summer School on Multi-Rate Processes, Slow-Fast Systems and Hysteresis MURPHYS-HSFS-2017, June 19 - 23, 2017, DISMA Politecnico di Torino, Dipartimento di Scienze Matematiche ``Giuseppe Luigi Lagrange'', Italy.

      • CH. Merdon, Pressure-robust mixed finite element methods for the Navier--Stokes equations, scMatheon Workshop RMMM 8 - Berlin 2017, Reliable Methods of Mathematical Modeling, July 31 - August 3, 2017, Humboldt-Universität zu Berlin, August 2, 2017.