Model based investigations of electrochemical double layers, porous catalysts in fuel cells, and battery materials
The behavior of electrochemical systems is widely investigated with continuum physics models. Applications range from single crystal electrochemistry to lithium batteries and fuel cells, from biological nanopores to electrolysis and corrosion science, and further. The common basis for all application is the theory of nonequilibrium thermoelectrodynamics [1,2]. At the WeierstraßInstitute general models for electrochemical systems are systematically derived. Asymptotic analysis methods are further employed to derived reduced nonequilibrium models for various electrodes and electrolytes, and the corresponding boundary conditions.Metal/electrolyte interface
A detailed model for a general electrolytic mixture was developed, which may adsorb and react on a metal surface. It is based on coupled volume and surface thermodynamics [3] where we account for adsorption and solvation of the ionic species.
Figure 1: Computed structure and a resulting sketch of the Ag/0.1M NaF interface.
Figure 2: Computed capacity of the Ag(110) 0.1M NaF interface.
Electron transfer reactions
Electron transfer reactions at the interface between an electrolyte and an electrode are the pivotal phenomenon of all electrochemical systems for storage and conversion of energy. The ButlerVolmer equation describes the dependence of the reaction rates on a potential difference across the interface and on the concentrations of the different species at the interface. At WIAS, new boundary conditions of generalized ButlerVolmer type were derived based on nonequilibrium thermodynamics [5]. The predictive capabilities of this theory are validated for various, well defined electrochemical cells. and is applied to complex systems like batteries and fuel cells.
Figure 3: Experimental setup copper deposition.
Figure 4: CurrentVoltage diagram for the electrodeposition of metal with different electrolyte concentrations. At high imposed currents diffusional transport in the electrolyte causes a lack of reacting ions at the electrode, leading to a blowup of the potential.
Thermodynamically consistent discretizations
The numerical solution of generalized PoissonNernstPlanck systems like the one derived in [5] in higher space dimensions and general geometries requires the development of specifically tailored discretization approaches which have the potential to preserve the thermodynamic properties of the continuous problem. For this purpose, a generalization of the ScharfetterGummel upwind finite volume scheme successfully employed in the field of semiconductor device simulation to the case PoissonNernstPlanck problems with ion volume constraints and solvent balancing has been proposed [6].
Figure 5: Simulated IV curves for an electrolytic diode. Difference between standard and improved NernstPlanck models.
Modeling of transport and reaction processes for magnesiumair batteries
As magnesium is highly abundant, comparably cheap and sufficiently reactive, rechargeable magnesium air batteries are an interesting option for large scale energy storage. The development of strategies for the realization of this battery type is the subject of the research network MgLuft which is funded by the German Ministry of Education and Research. The WIAS subproject is concerned with the model based interpretation of flow cell experiments supporting the acquisition of transport data in organic electrolytes and information or reaction kinetics [7] and the modeling of transport and reaction processes in the electrodes of such a cell.
Figure 6: Calculated streamlines of electrolyte flow in an experimental flow cell.
References
[1]  I. Müller, Thermodynamics, Pitman, 1985. 
[2]  S. de Groot, P. Mazur, NonEquilibrium Thermodynamics, Dover Publications, 1984. 
[3]  W. Dreyer, C. Guhlke and M. Landstorfer, Theory and structure of the metal electrolyte/interface incorporating adsorption and solvation effects, Preprint no. 2058, WIAS, Berlin, 2014. 
[4]  W. Dreyer, C. Guhlke and M. Landstorfer, A mixture theory of electrolytes containing solvation effects, Electrochemistry Communications, 43 (2014), pp. 7578. 
[5]  W. Dreyer, C. Guhlke and R. Müller, Modeling of electrochemical double layers in thermodynamic non equilibrium, Phys. Chem. Chem. Phys., 17 (2015), pp. 2717627194. 
[6]  J. Fuhrmann, Comparison and numerical treatment of generalised NernstPlanck models, Computer Physics Communications, 196 (2015), pp. 166178. 
[7]  J. Fuhrmann, A. Linke, C. Merdon, F. Neumann, T. Streckenbach, H. Baltruschat, and M. Khodayari, Inverse modeling of thin layer flow cells for detection of solubility, transport and reaction coefficients from experimental data, Preprint no. 2161, WIAS, Berlin, 2015. 
Publications
Monographs

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

W. Dreyer, C. Guhlke, R. Müller, The impact of solvation and dissociation on the transport parameters of liquid electrolytes: Continuum modeling and numerical study, European Physical Journal Special Topics, 227 (2019), pp. 25152538, DOI 10.1140/epjst/e20198001332 .
Abstract
Electrothermodynamics 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 ionsolvent 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 MaxwellStefan setting and incorporate them into the general electrothermodynamic 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 nondilute electrolyte. 
W. Dreyer, P. Friz, P. Gajewski, C. Guhlke, M. Maurelli, Stochastic manyparticle model for LFP electrodes, Continuum Mechanics and Thermodynamics, 30 (2018), pp. 593628, DOI 10.1007/s0016101806297 .
Abstract
In the framework of nonequilibrium 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 lithiumpoor to a lithiumrich 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 voltagecurrent 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. 
W. Dreyer, C. Guhlke, R. Müller, Bulksurface electrothermodynamics and applications to electrochemistry, Entropy. An International and Interdisciplinary Journal of Entropy and Information Studies, 20 (2018), pp. 939/1939/44, DOI 10.3390/e20120939 .
Abstract
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. 
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. 708753, DOI 10.1017/S0956792517000341 .
Abstract
The Lippmann equation is considered as universal relationship between interfacial tension, double layer charge, and cell potential. Based on the framework of continuum thermoelectrodynamics 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 mercuryelectrolyte 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. Khodayari, P. Reinsberg, A.A. AbdElLatif, Ch. Merdon, J. Fuhrmann, H. Baltruschat, Determining solubility and diffusivity by using a flow cell coupled to a mass spectrometer, ChemPhysChem, 17 (2016), pp. 16471655.

W. Dreyer, C. Guhlke, M. Landstorfer, Theory and structure of the metal/electrolyte interface incorporating adsorption and solvation effects, Electrochimica Acta, 201 (2016), pp. 187219.
Abstract
In this work we present a continuum theory for the metal/electrolyte interface which explicitly takes into account adsorption and partial solvation on the metal surface. It is based on a general theory of coupled thermoelectrodynamics for volumes and surfaces, utilized here in equilibrium and a 1D approximation. We provide explicit free energy models for the volumetric metal and electrolyte phases and derive a surface free energy for the species present on the metal surface. This surface mixture theory explicitly takes into account the very different amount of sites an adsorbate requires, originating from solvation effects on the surface. Additionally we account for electron transfer reactions on the surface and the associated stripping of the solvation shell. Based on our overall surface free energy we thus provide explicit expressions of the surface chemical potentials of all constituents. The equilibrium representations of the coverages and the overall charge are briefly summarized.
Our model is then used to describe two examples: (i) a silver single crystal electrode with (100) face in contact to a (0.01M NaF + 0.01M KPF6) aqueous solution, and (ii) a general metal surface in contact to some electrolytic solution AC for which an electron transfer reaction occurs in the potential range of interest. We reflect the actual modeling procedure for these examples and discuss the respective model parameters. Due to the representations of the coverages in terms of the applied potential we provide an adsorption map and introduce adsorption potentials. Finally we investigate the structure of the space charge layer at the metal/surface/electrolyte interface by means of numerical solutions of the coupled Poissonmomentum equation system for various applied potentials. It turns out that various layers selfconsistently form within the overall space charge region, which are compared to historic and recent pictures of the double layer. Based on this we present new interpretations of what is known as inner and outer Helmholtzplanes and finally provide a thermodynamic consistent picture of the metal/electrolyte interface structure. 
W. Dreyer, C. Guhlke, R. Müller, A new perspective on the electron transfer: Recovering the ButlerVolmer equation in nonequilibrium thermodynamics, Physical Chemistry Chemical Physics, 18 (2016), pp. 2496624983, DOI 10.1039/C6CP04142F .
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 ButlerVolmer equation which has its origin in kinetic theories. The ButlerVolmer equation relates interfacial reaction rates to bulk quantities like the electrostatic potential and electrolyte concentrations. Since in the classical form, the validity of the ButlerVolmer equation is limited to some simple electrochemical systems, many attempts have been made to generalize the ButlerVolmer equation. Based on nonequilibrium thermodynamics we have recently derived a reduced model for the electrodeelectrolyte 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 ButlerVolmer equation. We discuss the application of the new ButlerVolmer equations to different scenarios like electron transfer reactions at metal electrodes, the intercalation process in lithiumironphosphate electrodes and adsorption processes. We illustrate the theory by an example of electroplating. 
J. Fuhrmann, A numerical strategy for NernstPlanck systems with solvation effect, Fuel Cells, 16 (2016), pp. 704714.

J. Fuhrmann, A. Linke, Ch. Merdon, F. Neumann, T. Streckenbach, H. Baltruschat, M. Khodayari, Inverse modeling of thin layer flow cells for detection of solubility, transport and reaction coefficients from experimental data, Electrochimica Acta, 211 (2016), pp. 110.
Abstract
Thin layer flow cells are used in electrochemical research as experimental devices which allow to perform investigations of electrocatalytic surface reactions under controlled conditions using reasonably small electrolyte volumes. The paper introduces a general approach to simulate the complete cell using accurate numerical simulation of the coupled flow, transport and reaction processes in a flow cell. The approach is based on a mass conservative coupling of a divergencefree finite element method for fluid flow and a stable finite volume method for mass transport. It allows to perform stable and efficient forward simulations that comply with the physical bounds namely mass conservation and maximum principles for the involved species. In this context, several recent approaches to obtain divergencefree velocities from finite element simulations are discussed. In order to perform parameter identification, the forward simulation method is coupled to standard optimization tools. After an assessment of the inverse modeling approach using known realistic data, first results of the identification of solubility and transport data for O2 dissolved in organic electrolytes are presented. A plausibility study for a more complex situation with surface reactions concludes the paper and shows possible extensions of the scope of the presented numerical tools. 
W. Dreyer, C. Guhlke, R. Müller, Modeling of electrochemical double layers in thermodynamic nonequilibrium, Physical Chemistry Chemical Physics, 17 (2015), pp. 2717627194, DOI 10.1039/C5CP03836G .
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 metalelectrolyte interface, we derive a qualitative description of the double layer capacitance without the need to resolve space charge layers. 
J. Fuhrmann, Comparison and numerical treatment of generalized NernstPlanck models, Computer Physics Communications. An International Journal and Program Library for Computational Physics and Physical Chemistry, 196 (2015), pp. 166178.
Abstract
In its most widespread, classical formulation, the NernstPlanckPoisson system for ion transport in electrolytes fails to take into account finite ion sizes. As a consequence, it predicts unphysically high ion concentrations near electrode surfaces. Historical and recent approaches to an approriate modification of the model are able to fix this problem. Several appropriate formulations are compared in this paper. The resulting equations are reformulated using absolute activities as basic variables describing the species amounts. This reformulation allows to introduce a straightforward generalisation of the ScharfetterGummel finite volume discretization scheme for driftdiffusion equations. It is shown that it is thermodynamically consistent in the sense that the solution of the corresponding discretized generalized PoissonBoltzmann system describing the thermodynamic equilibrium is a stationary state of the discretized timedependent generalized NernstPlanck system. Numerical examples demonstrate the improved physical correctness of the generalised models and the feasibility of the numerical approach. 
A. Mielke, J. Haskovec, P.A. Markowich, On uniform decay of the entropy for reactiondiffusion systems, Journal of Dynamics and Differential Equations, 27 (2015), pp. 897928.
Abstract
In this work we derive entropy decay estimates for a class of nonlinear reactiondiffusion systems modeling reversible chemical reactions under the assumption of detailed balance. In particular, we provide explicit bounds for the exponential decay of the relative logarithmic entropy, being based essentially on the application of the logSobolev inequality and a convexification argument only, making it quite robust to model variations. An important feature of our analysis is the interaction of the two different dissipative mechanisms: pure diffusion, forcing the system asymptotically to the homogeneous state, and pure reaction, forcing the solution to the (possibly inhomogeneous) chemical equilibrium. Only the interaction of both mechanisms provides the convergence to the homogeneous equilibrium. Moreover, we introduce two generalizations of the main result: we allow for vanishing diffusion constants in some chemical components, and we consider different entropy functionals. We provide a few examples to highlight the usability of our approach and shortly discuss possible further applications and open questions. 
M. Liero, A. Mielke, Gradient structures and geodesic convexity for reactiondiffusion systems, Philosophical Transactions of the Royal Society A : Mathematical, Physical & Engineering Sciences, 371 (2013), pp. 20120346/120120346/28.
Abstract
We consider systems of reactiondiffusion equations as gradient systems with respect to an entropy functional and a dissipation metric given in terms of a socalled Onsager operator, which is a sum of a diffusion part of Wasserstein type and a reaction part. We provide methods for establishing geodesic lambdaconvexity of the entropy functional by purely differential methods, thus circumventing arguments from mass transportation. Finally, several examples, including a driftdiffusion system, provide a survey on the applicability of the theory. We consider systems of reactiondiffusion equations as gradient systems with respect to an entropy functional and a dissipation metric given in terms of a socalled Onsager operator, which is a sum of a diffusion part of Wasserstein type and a reaction part. We provide methods for establishing geodesic lambdaconvexity of the entropy functional by purely differential methods, thus circumventing arguments from mass transportation. Finally, several examples, including a driftdiffusion system, provide a survey on the applicability of the theory. 
M. Liero, Passing from bulk to bulk/surface evolution in the AllenCahn equation, NoDEA. Nonlinear Differential Equations and Applications, 20 (2013), pp. 919942.
Abstract
In this paper we formulate a boundary layer approximation for an AllenCahntype equation involving a small parameter $eps$. Here, $eps$ is related to the thickness of the boundary layer and we are interested in the limit when $eps$ tends to 0 in order to derive nontrivial boundary conditions. The evolution of the system is written as an energy balance formulation of the L^2gradient flow with the corresponding AllenCahn energy functional. By transforming the boundary layer to a fixed domain we show the convergence of the solutions to a solution of a limit system. This is done by using concepts related to Gamma and Mosco convergence. By considering different scalings in the boundary layer we obtain different boundary conditions. 
A. Glitzky, A. Mielke, A gradient structure for systems coupling reactiondiffusion effects in bulk and interfaces, ZAMP Zeitschrift fur Angewandte Mathematik und Physik. ZAMP. Journal of Applied Mathematics and Physics. Journal de Mathematiques et de Physique Appliquees, 64 (2013), pp. 2952.
Abstract
We derive gradientflow formulations for systems describing driftdiffusion processes of a finite number of species which undergo massaction type reversible reactions. Our investigations cover heterostructures, where material parameter may depend in a nonsmooth way on the space variable. The main results concern a gradient flow formulation for electroreactiondiffusion systems with active interfaces permitting driftdiffusion processes and reactions of species living on the interface and transfer mechanisms allowing bulk species to jump into an interface or to pass through interfaces. The gradient flows are formulated in terms of two functionals: the free energy and the dissipation potential. Both functionals consist of a bulk and an interface integral. The interface integrals determine the interface dynamics as well as the selfconsistent coupling to the model in the bulk. The advantage of the gradient structure is that it automatically generates thermodynamically consistent models. 
W. Dreyer, C. Guhlke, R. Müller, Overcoming the shortcomings of the NernstPlanck model, Physical Chemistry Chemical Physics, 15 (2013), pp. 70757086, DOI 10.1039/C3CP44390F .
Abstract
This is a study on electrolytes that takes a thermodynamically consistent coupling between mechanics and diffusion into account. It removes some inherent deficiencies of the popular NernstPlanck model. A boundary problem for equilibrium processes is used to illustrate the new features of our model. 
A. Mielke, Thermomechanical modeling of energyreactiondiffusion systems, including bulkinterface interactions, Discrete and Continuous Dynamical Systems  Series S, 6 (2013), pp. 479499.
Abstract
We show that many couplings between parabolic systems for processes in solids can be formulated as a gradient system with respect to the total free energy or the total entropy. This includes AllenCahn, CahnHilliard, and reactiondiffusion systems and the heat equation. For this, we write the coupled system as an Onsager system (X,Φ,K) defining the evolution $dot U$=  K(U) DΦ(U). Here Φ is the driving functional, while the Onsager operator K(U) is symmetric and positive semidefinite. If the inverse G=K^{1} exists, the triple (X,Φ,G) defines a gradient system. Onsager systems are well suited to model bulkinterface interactions by using the dual dissipation potential Ψ^{*}(U, Ξ)= ½ ⟨Ξ K(U) Ξ⟩. Then, the two functionals Φ and Ψ^{*} can be written as a sum of a volume integral and a surface integral, respectively. The latter may contain interactions of the driving forces in the interface as well as the traces of the driving forces from the bulk. Thus, capture and escape mechanisms like thermionic emission appear naturally in Onsager systems, namely simply through integration by parts. 
M. Liero, Th. Roche, Rigorous derivation of a plate theory in linear elastoplasticity via $Gamma$convergence, NoDEA. Nonlinear Differential Equations and Applications, 19 (2012), pp. 437457.
Abstract
This paper deals with dimension reduction in linearized elastoplasticity in the rateindependent case. The reference configuration of the elastoplastic body is given by a twodimensional middle surface and a small but positive thickness. We derive a limiting model for the case in which the thickness of the plate tends to 0. This model contains membrane and plate deformations which are coupled via plastic strains. The convergence analysis is based on an abstract Gamma convergence theory for rateindependent evolution formulated in the framework of energetic solutions. This concept is based on an energystorage functional and a dissipation functional, such that the notion of solution is phrased in terms of a stability condition and an energy balance. 
CH. Batallion, F. Bouchon, C. ChainaisHillairet, J. Fuhrmann, E. Hoarau, R. Touzani, Numerical methods for the simulation of a corrosion model in a nuclear waste deep repository, Journal of Computational Physics, 231 (2012), pp. 62136231.
Abstract
In this paper, we design numerical methods for a PDE system arising in corrosion modelling. This system describes the evolution of a dense oxide layer. It is based on a driftdiffusion system and includes moving boundary equations. The choice of the numerical methods is justified by a stability analysis and by the study of their numerical performance. Finally, numerical experiments with reallife data shows the efficiency of the developed methods. 
A. Glitzky, An electronic model for solar cells including active interfaces and energy resolved defect densities, SIAM Journal on Mathematical Analysis, 44 (2012), pp. 38743900.
Abstract
We introduce an electronic model for solar cells taking into account heterostructures with active interfaces and energy resolved volume and interface trap densities. The model consists of continuity equations for electrons and holes with thermionic emission transfer conditions at the interface and of ODEs for the trap densities with energy level and spatial position as parameters, where the right hand sides contain generationrecombination as well as ionization reactions. This system is coupled with a Poisson equation for the electrostatic potential. We show the thermodynamic correctness of the model and prove a priori estimates for the solutions to the evolution system. Moreover, existence and uniqueness of weak solutions of the problem are proven. For this purpose we solve a regularized problem and verify bounds of the corresponding solution not depending on the regularization level. 
J. Fuhrmann, H. Zhao, H. Langmach, Y.E. Seidel, Z. Jusys, R.J. Behm, The role of reactive reaction intermediates in twostep heterogeneous electrocatalytic reactions: A model study, Fuel Cells, 11 (2011), pp. 501510.
Abstract
Experimental investigations of heterogeneous electrocatalytic reactions have been performed in flow cells which provide an environment with controlled parameters. Measurements of the oxygen reduction reaction in a flow cell with an electrode consisting of an array of Pt nanodisks on a glassy carbon substrate exhibited a decreasing fraction of the intermediate $H_2O_2$ in the overall reaction products with increasing density of the nanodiscs. A similar result is true for the dependence on the catalyst loading in the case of a supported Pt/C catalyst thinfilm electrode, where the fraction of the intermediate decreases with increasing catalyst loading. Similar effects have been detected for the methanol oxidation. We present a model of multistep heterogeneous electrocatalytic oxidation and reduction reactions based on an adsorptionreactiondesorption scheme using the Langmuir assumption and macroscopic transport equations. A continuum based model problem in a vertical cross section of a rectangular flow cell is proposed in order to explain basic principles of the experimental situation. It includes three model species A, B, C, which undergo adsorption and desorption at a catalyst surface, as well as adsorbate reactions from A to B to C. These surface reactions are coupled with diffusion and advection in the Hagen Poiseuille flow in the flow chamber of the cell. Both high velocity asymptotic theory and a finite volume numerical are used to obtain approximate solutions to the model. Both approaches show a behaviour similar to the experimentally observed. Working in more general situations, the finite volume scheme was applied to a catalyst layer consisting of a number of small catalytically active areas corresponding to nanodisks. Good qualitative agreement with the experimental findings was established for this case as well. 
A. Mielke, A gradient structure for reactiondiffusion systems and for energydriftdiffusion systems, Nonlinearity, 24 (2011), pp. 13291346.
Abstract
In recent years the theory of Wasserstein distances has opened up a new treatment of the diffusion equations as gradient systems, where the entropy takes the role of the driving functional and where the space is equipped with the Wasserstein metric. We show that this structure can be generalized to closed reactiondiffusion systems, where the free energy (or the entropy) is the driving functional and further conserved quantities may exists, like the total number of chemical species. The metric is constructed by using the dual dissipation potential, which is a convex function of the chemical potentials. In particular, it is possible to treat diffusion and reaction terms simultaneously. The same ideas extend to semiconductor equations involving the electron and hole densities, the electrostatic potential, and the temperature. 
A. Glitzky, Uniform exponential decay of the free energy for Voronoi finite volume discretized reactiondiffusion systems, Mathematische Nachrichten, 284 (2011), pp. 21592174.
Abstract
Our focus are energy estimates for discretized reactiondiffusion systems for a finite number of species. We introduce a discretization scheme (Voronoi finite volume in space and fully implicit in time) which has the special property that it preserves the main features of the continuous systems, namely positivity, dissipativity and flux conservation. For a class of Voronoi finite volume meshes we investigate thermodynamic equilibria and prove for solutions to the evolution system the monotone and exponential decay of the discrete free energy to its equilibrium value with a unified rate of decay for this class of discretizations. The essential idea is an estimate of the free energy by the dissipation rate which is proved indirectly by taking into account sequences of Voronoi finite volume meshes. Essential ingredient in that proof is a discrete SobolevPoincaré inequality. 
A. Glitzky, J.A. Griepentrog, Discrete SobolevPoincaré inequalities for Voronoi finite volume approximations, SIAM Journal on Numerical Analysis, 48 (2010), pp. 372391.
Abstract
We prove a discrete SobolevPoincare inequality for functions with arbitrary boundary values on Voronoi finite volume meshes. We use Sobolev's integral representation and estimate weakly singular integrals in the context of finite volumes. We establish the result for star shaped polyhedral domains and generalize it to the finite union of overlapping star shaped domains. In the appendix we prove a discrete Poincare inequality for space dimensions greater or equal to two. 
R. HallerDintelmann, Ch. Meyer, J. Rehberg, A. Schiela, Hölder continuity and optimal control for nonsmooth elliptic problems, Applied Mathematics and Optimization. An International Journal with Applications to Stochastics, 60 (2009), pp. 397428.
Abstract
The well known De Giorgi result on Hölder continuity for solutions of the Dirichlet problem is reestablished for mixed boundary value problems, provided that the underlying domain is a Lipschitz domain and the border between the Dirichlet and the Neumann boundary part satisfies a very general geometric condition. Implications of this result for optimal control theory are presented. 
R. HallerDintelmann, J. Rehberg, Maximal parabolic regularity for divergence operators including mixed boundary conditions, Journal of Differential Equations, 247 (2009), pp. 13541396.
Abstract
We show that elliptic second order operators $A$ of divergence type fulfill maximal parabolic regularity on distribution spaces, even if the underlying domain is highly nonsmooth and $A$ is complemented with mixed boundary conditions. Applications to quasilinear parabolic equations with nonsmooth data are presented. 
J. Fuhrmann, A. Linke, H. Langmach, H. Baltruschat, Numerical calculation of the limiting current for a cylindrical thin layer flow cell, Electrochimica Acta, 55 (2009), pp. 430438.

A. Glitzky, K. Gärtner, Energy estimates for continuous and discretized electroreactiondiffusion systems, Nonlinear Analysis. Theory, Methods & Applications. An International Multidisciplinary Journal. Series A: Theory and Methods, 70 (2009), pp. 788805.
Abstract
We consider electroreactiondiffusion systems consisting of continuity equations for a finite number of species coupled with a Poisson equation. We take into account heterostructures, anisotropic materials and rather general statistic relations.
We investigate thermodynamic equilibria and prove for solutions to the evolution system the monotone and exponential decay of the free energy to its equilibrium value. Here the essential idea is an estimate of the free energy by the dissipation rate which is proved indirectly.
The same properties are shown for an implicit time discretized version of the problem. Moreover, we provide a space discretized scheme for the electroreactiondiffusion system which is dissipative (the free energy decays monotonously). On a fixed grid we use for each species different Voronoi boxes which are defined with respect to the anisotropy matrix occurring in the flux term of this species. 
A. Glitzky, Energy estimates for electroreactiondiffusion systems with partly fast kinetics, Discrete and Continuous Dynamical Systems, 25 (2009), pp. 159174.
Abstract
We start from a basic model for the transport of charged species in heterostructures containing the mechanisms diffusion, drift and reactions in the domain and at its boundary. Considering limit cases of partly fast kinetics we derive reduced models. This reduction can be interpreted as some kind of projection scheme for the weak formulation of the basic electroreactiondiffusion system. We verify assertions concerning invariants and steady states and prove the monotone and exponential decay of the free energy along solutions to the reduced problem and to its fully implicit discretetime version by means of the results of the basic problem. Moreover we make a comparison of prolongated quantities with the solutions to the basic model. 
R. HallerDintelmann, H.Chr. Kaiser, J. Rehberg, Elliptic model problems including mixed boundary conditions and material heterogeneities, Journal de Mathématiques Pures et Appliquées, 89 (2008), pp. 2548.

M. Hieber, J. Rehberg, Quasilinear parabolic systems with mixed boundary conditions on nonsmooth domains, SIAM Journal on Mathematical Analysis, 40 (2008), pp. 292305.
Abstract
In this paper we investigate quasilinear systems of reactiondiffusion equations with mixed DirichletNeumann bondary conditions on non smooth domains. Using techniques from maximal regularity and heatkernel estimates we prove existence of a unique solution to systems of this type. 
J. Fuhrmann, H. Zhao, E. Holzbecher, H. Langmach, Flow, transport, and reactions in a thin layer flow cell, Journal of Fuel Cell Science and Technology, 5 (2008), pp. 021008/1021008/10.

A. Glitzky, Exponential decay of the free energy for discretized electroreactiondiffusion systems, Nonlinearity, 21 (2008), pp. 19892009.
Abstract
Our focus are electroreactiondiffusion systems consisting of continuity equations for a finite number of species coupled with a Poisson equation. We take into account heterostructures, anisotropic materials and rather general statistical relations. We introduce a discretization scheme (in space and fully implicit in time) using a fixed grid but for each species different Voronoi boxes which are defined with respect to the anisotropy matrix occurring in the flux term of this species. This scheme has the special property that it preserves the main features of the continuous systems, namely positivity, dissipativity and flux conservation. For the discretized electroreactiondiffusion system we investigate thermodynamic equilibria and prove for solutions to the evolution system the monotone and exponential decay of the free energy to its equilibrium value. The essential idea is an estimate of the free energy by the dissipation rate which is proved indirectly. 
J.A. Griepentrog, Maximal regularity for nonsmooth parabolic problems in SobolevMorrey spaces, Advances in Differential Equations, 12 (2007), pp. 10311078.
Abstract
This text is devoted to maximal regularity results for second order parabolic systems on LIPSCHITZ domains of space dimension greater or equal than three with diagonal principal part, nonsmooth coefficients, and nonhomogeneous mixed boundary conditions. We show that the corresponding class of initial boundary value problems generates isomorphisms between two scales of SOBOLEVMORREY spaces for solutions and right hand sides introduced in the first part of our presentation. The solutions depend smoothly on the data of the problem. Moreover, they are HOELDER continuous in time and space up to the boundary for a certain range of MORREY exponents. Due to the complete continuity of embedding and trace maps these results remain true for a broad class of unbounded lower order coefficients. 
J.A. Griepentrog, SobolevMorrey spaces associated with evolution equations, Advances in Differential Equations, 12 (2007), pp. 781840.
Abstract
In this text we introduce new classes of SOBOLEVMORREY spaces being adequate for the regularity theory of second order parabolic boundary value problems on LIPSCHITZ domains of space dimension greater or equal than three with nonsmooth coefficients and mixed boundary conditions. We prove embedding and trace theorems as well as invariance properties of these spaces with respect to localization, LIPSCHITZ transformation, and reflection. In the second part of our presentation we show that the class of second order parabolic systems with diagonal principal part generates isomorphisms between the above mentioned SOBOLEVMORREY spaces of solutions and right hand sides. 
O. Minet, H. Gajewski, J.A. Griepentrog, J. Beuthan, The analysis of laser light scattering during rheumatoid arthritis by image segmentation, Laser Physics Letters, 4 (2007), pp. 604610.

J. Elschner, H.Chr. Kaiser, J. Rehberg, G. Schmidt, $W^1,q$ regularity results for elliptic transmission problems on heterogeneous polyhedra, Mathematical Models & Methods in Applied Sciences, 17 (2007), pp. 593615.

J. Elschner, J. Rehberg, G. Schmidt, Optimal regularity for elliptic transmission problems including $C^1$ interfaces, Interfaces and Free Boundaries. Mathematical Modelling, Analysis and Computation, 9 (2007), pp. 233252.
Abstract
We prove an optimal regularity result for elliptic operators $nabla cdot mu nabla:W^1,q_0 rightarrow W^1,q$ for a $q>3$ in the case when the coefficient function $mu$ has a jump across a $C^1$ interface and is continuous elsewhere. A counterexample shows that the $C^1$ condition cannot be relaxed in general. Finally, we draw some conclusions for corresponding parabolic operators. 
A. Glitzky, R. Hünlich, Resolvent estimates in $W^1,p$ related to strongly coupled linear parabolic systems with coupled nonsmooth capacities, Mathematical Methods in the Applied Sciences, 30 (2007), pp. 22152232.
Abstract
We investigate linear parabolic systems with coupled nonsmooth capacities and mixed boundary conditions. We prove generalized resolvent estimates in $W^1,p$ spaces. The method is an appropriate modification of a technique introduced by Agmon to obtain $L^p$ estimates for resolvents of elliptic differential operators in the case of smooth boundary conditions. Moreover, we establish an existence and uniqueness result. 
H. Gajewski, J.A. Griepentrog, A descent method for the free energy of multicomponent systems, Discrete and Continuous Dynamical Systems, 15 (2006), pp. 505528.

H.Chr. Kaiser, H. Neidhardt, J. Rehberg, Classical solutions of quasilinear parabolic systems on two dimensional domains, NoDEA. Nonlinear Differential Equations and Applications, 13 (2006), pp. 287310.

M. Baro, H. Neidhardt, J. Rehberg, Current coupling of driftdiffusion models and dissipative SchrödingerPoisson systems: Dissipative hybrid models, SIAM Journal on Mathematical Analysis, 37 (2005), pp. 941981.

A. Glitzky, R. Hünlich, Global existence result for pair diffusion models, SIAM Journal on Mathematical Analysis, 36 (2005), pp. 12001225.

A. Glitzky, R. Hünlich, Stationary energy models for semiconductor devices with incompletely ionized impurities, ZAMM. Zeitschrift für Angewandte Mathematik und Mechanik, 85 (2005), pp. 778792.

J. Rehberg, Quasilinear parabolic equations in $L^p$, Progress in Nonlinear Differential Equations and their Applications, 64 (2005), pp. 413419.

V. Maz'ya, J. Elschner, J. Rehberg, G. Schmidt, Solutions for quasilinear nonsmooth evolution systems in $L^p$, Archive for Rational Mechanics and Analysis, 171 (2004), pp. 219262.

H. Gajewski, I.V. Skrypnik, On unique solvability of nonlocal driftdiffusiontype problems, Nonlinear Analysis. Theory, Methods & Applications. An International Multidisciplinary Journal. Series A: Theory and Methods, 56 (2004), pp. 803830.

H. Gajewski, I.V. Skrypnik, To the uniqueness problem for nonlinear parabolic equations, Discrete and Continuous Dynamical Systems, 10 (2004), pp. 315336.

A. Glitzky, W. Merz, Single dopant diffusion in semiconductor technology, Mathematical Methods in the Applied Sciences, 27 (2004), pp. 133154.

A. Glitzky, R. Hünlich, Stationary solutions of twodimensional heterogeneous energy models with multiple species, Banach Center Publications, 66 (2004), pp. 135151.

A. Glitzky, Electroreactiondiffusion systems with nonlocal constraints, Mathematische Nachrichten, 277 (2004), pp. 1446.

H. Gajewski, K. Zacharias, On a nonlocal phase separation model, Journal of Mathematical Analysis and Applications, 286 (2003), pp. 1131.

G. Albinus, H. Gajewski, R. Hünlich, Thermodynamic design of energy models of semiconductor devices, Nonlinearity, 15 (2002), pp. 367383.

H. Gajewski, On a nonlocal model of nonisothermal phase separation, Advances in Mathematical Sciences and Applications, 12 (2002), pp. 569586.

J.A. Griepentrog, K. Gröger, H.Chr. Kaiser, J. Rehberg, Interpolation for function spaces related to mixed boundary value problems, Mathematische Nachrichten, 241 (2002), pp. 110120.

J.A. Griepentrog, Linear elliptic boundary value problems with nonsmooth data: Campanato spaces of functionals, Mathematische Nachrichten, 243 (2002), pp. 1942.

A. Glitzky, R. Hünlich, Global properties of pair diffusion models, Advances in Mathematical Sciences and Applications, 11 (2001), pp. 293321.

J.A. Griepentrog, H.Chr. Kaiser, J. Rehberg, Heat kernel and resolvent properties for second order elliptic differential operators with general boundary conditions on $Lsp p$, Advances in Mathematical Sciences and Applications, 11 (2001), pp. 87112.

W. Merz, A. Glitzky, R. Hünlich, K. Pulverer, Strong solutions for pair diffusion models in homogeneous semiconductors, Nonlinear Analysis. Real World Applications. An International Multidisciplinary Journal, 2 (2001), pp. 541567.

J.A. Griepentrog, L. Recke, Linear elliptic boundary value problems with nonsmooth data: Normal solvability on SobolevCampanato spaces, Mathematische Nachrichten, 225 (2001), pp. 3974.

A. Glitzky, R. Hünlich, Electroreactiondiffusion systems including cluster reactions of higher order, Mathematische Nachrichten, 216 (2000), pp. 95118.
Contributions to Collected Editions

A. Fiebach, A. Glitzky, Uniform estimate of the relative free energy by the dissipation rate for finite volume discretized reactiondiffusion systems, in: Finite Volumes for Complex Applications VII  Methods and Theoretical Aspects  FVCA 7, Berlin, June 2014, J. Fuhrmann, M. Ohlberger, Ch. Rohde, eds., 77 of Springer Proceedings in Mathematics & Statistics, Springer International Publishing, Cham et al., 2014, pp. 275283.
Abstract
We prove a uniform Poincarelike estimate of the relative free energy by the dissipation rate for implicit Euler, finite volume discretized reactiondiffusion systems. This result is proven indirectly and ensures the exponential decay of the relative free energy with a unified decay rate for admissible finite volume meshes. 
A. Glitzky, A. Mielke, L. Recke, M. Wolfrum, S. Yanchuk, D2  Mathematics for optoelectronic devices, in: MATHEON  Mathematics for Key Technologies, M. Grötschel, D. Hömberg, J. Sprekels, V. Mehrmann ET AL., eds., 1 of EMS Series in Industrial and Applied Mathematics, European Mathematical Society Publishing House, Zurich, 2014, pp. 243256.

J. Fuhrmann, A. Linke, Ch. Merdon, Coupling of fluid flow and solute transport using a divergencefree reconstruction of the CrouzeixRaviart element, in: Finite Volumes for Complex Applications VII  Elliptic, Parabolic and Hyperbolic Problems  FVCA 7, Berlin, June 2014, J. Fuhrmann, M. Ohlberger, Ch. Rohde, eds., 78 of Springer Proceedings in Mathematics & Statistics, Springer International Publishing, Cham et al., 2014, pp. 587595.

J. Fuhrmann, K. Gärtner, Modeling of twophase flow and catalytic reaction kinetics for DMFCs, in: Device and Materials Modeling in PEM Fuel Cells, S. Paddison, K. Promislow, eds., 113 of Topics in Applied Physics, Springer, Berlin/Heidelberg, 2009, pp. 297316.

H. Gajewski, J.A. Griepentrog, A. Mielke, J. Beuthan, U. Zabarylo, O. Minet, Image segmentation for the investigation of scatteredlight images when laseroptically diagnosing rheumatoid arthritis, in: Mathematics  Key Technology for the Future, W. Jäger, H.J. Krebs, eds., Springer, Heidelberg, 2008, pp. 149161.

M. Ehrhardt, J. Fuhrmann, A. Linke, E. Holzbecher, Mathematical modeling of channelporous layer interfaces in PEM fuel cells, in: Proceedings of FDFC2008  Fundamentals and Developments of Fuel Cell Conference 2008, Nancy, France, December 1012 (CD), 2008, pp. 8 pages.
Abstract
In proton exchange membrane (PEM) fuel cells, the transport of the fuel to the active zones, and the removal of the reaction products are realized using a combination of channels and porous diffusion layers. In order to improve existing mathematical and numerical models of PEM fuel cells, a deeper understanding of the coupling of the flow processes in the channels and diffusion layers is necessary.
After discussing different mathematical models for PEM fuel cells, the work will focus on the description of the coupling of the free flow in the channel region with the filtration velocity in the porous diffusion layer as well as interface conditions between them.
The difficulty in finding effective coupling conditions at the interface between the channel flow and the membrane lies in the fact that often the orders of the corresponding differential operators are different, e.g., when using stationary (Navier)Stokes and Darcy's equation. Alternatively, using the Brinkman model for the porous media this difficulty does not occur.
We will review different interface conditions, including the wellknown BeaversJosephSaffman boundary condition and its recent improvement by Le Bars and Worster. 
U. Bandelow, H. Gajewski, R. Hünlich, Thermodynamic designed energy model, in: Proceedings of the IEEE/LEOS 3rd International Conference on Numerical Simulation of Semiconductor Optoelectronic Devices (NUSOD'03), J. Piprek, ed., 2003, pp. 3537.

H. Gajewski, H.Chr. Kaiser, H. Langmach, R. Nürnberg, R.H. Richter, Mathematical modelling and numerical simulation of semiconductor detectors, in: Mathematics  Key Technology for the Future. Joint Projects Between Universities and Industry, W. Jäger, H.J. Krebs, eds., Springer, Berlin [u.a.], 2003, pp. 355364.

I.V. Skrypnik, H. Gajewski, On the uniqueness of solutions to nonlinear elliptic and parabolic problems (in Russian), in: Differ. Uravn. i Din. Sist., dedicated to the 80th anniversary of the Academician Evgenii Frolovich Mishchenko, Suzdal, 2000, 236 of Tr. Mat. Inst. Steklova, Moscow, Russia, 2002, pp. 318327.

U. Bandelow, H. Gajewski, H.Chr. Kaiser, Modeling combined effects of carrier injection, photon dynamics and heating in Strained MultiQuantumWell Laser, in: Physics and Simulation of Optoelectronic Devices VIII, R.H. Binder, P. Blood, M. Osinski, eds., 3944 of Proceedings of SPIE, SPIE, Bellingham, WA, 2000, pp. 301310.

G. Schwarz, E. Schöll, R. Nürnberg, H. Gajewski, Simulation of current filamentation in an extended driftdiffusion model, in: EQUADIFF 99: International Conference on Differential Equations, Berlin 1999, B. Fiedler, K. Gröger, J. Sprekels, eds., 2, World Scientific, Singapore [u. a.], 2000, pp. 13341336.

H. Gajewski, K. Zacharias, On a reactiondiffusion system modelling chemotaxis, in: EQUADIFF 99: International Conference on Differential Equations, Berlin 1999, B. Fiedler, K. Gröger, J. Sprekels, eds., 2, World Scientific, Singapore [u. a.], 2000, pp. 10981103.

H.Chr. Kaiser, J. Rehberg, About some mathematical questions concerning the embedding of SchrödingerPoisson systems into the driftdiffusion model of semiconductor devices, in: EQUADIFF 99: International Conference on Differential Equations, Berlin 1999, B. Fiedler, K. Gröger, J. Sprekels, eds., 2, World Scientific, Singapore [u. a.], 2000, pp. 13281333.
Preprints, Reports, Technical Reports

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 NernstPlanckPoisson driftdiffusion equations for ion transport and the Stokes resp. NavierStokes 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. 
L. Donati, M. Heida, M. Weber, B. Keller, Estimation of the infinitesimal generator by squareroot 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 timescales, 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 lowdimensional 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 2ddiffusion process and Alanine dipeptide. 
W. Dreyer, P.É. Druet, P. Gajewski, C. Guhlke, Analysis of improved NernstPlanckPoisson 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 NernstPlanckPoisson 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 convectiondiffusionreaction equations for the constituents of the mixture, of the NavierStokes equation for the barycentric velocity, and of the Poisson equation for the electrical potential. Due to the principle of mass conservation, crossdiffusion 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 AubinLions 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 NernstPlanckPoisson 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 NernstPlanckPoisson 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 convectiondiffusionreaction equations for the constituents of the mixture, of the NavierStokes equation for the barycentric velocity, and of the Poisson equation for the electrical potential. Due to the principle of mass conservation, crossdiffusion 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 NernstPlanckPoisson 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 NernstPlanckPoisson 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 convectiondiffusionreaction equations for the constituents of the mixture, of the NavierStokes equation for the barycentric velocity, and of the Poisson equation for the electrical potential. Due to the principle of mass conservation, crossdiffusion phenomena must occur and the mobility matrix (Onsager matrix) has a kernel. In this paper we establish the existence of a globalintime 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.
Talks, Poster

W. Dreyer, J. Fuhrmann, P. Gajewski, C. Guhlke, M. Landstorfer, M. Maurelli, R. Müller, Stochastic model for LiFePO4electrodes, ModVal14  14th Symposium on Fuel Cell and Battery Modeling and Experimental Validation, Karlsruhe, March 2  3, 2017.

CH. Merdon, A novel concept for the discretisation of the coupled NernstPlanckPoissonNavierStokes system, 14th Symposium on Fuel Cell Modelling and Experimental Validation (MODVAL 14), March 2  3, 2017, Karlsruher Institut für Technologie, Institut für Angewandte Materialien, Karlsruhe, Germany, March 3, 2017.

P.É. Druet, Analysis of recent NernstPlanckPoissonNavierStokes systems of electrolytes, 88th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2017), Section S14 ``Applied Analysis'', March 6  10, 2017, Bauhaus Universität Weimar/Technische Universität Ilmenau, Weimar, March 7, 2017.

P.É. Druet, Existence of weak solutions for improved NernstPlanckPoisson models of compressible electrolytes, Seminar EDE, Czech Academy of Sciences, Institute of Mathematics, Department of Evolution Differential Equations (EDE), Prague, Czech Republic, January 10, 2017.

CH. Merdon, J. Fuhrmann, A. Linke, A.A. AbdElLatif, M. Khodayari, P. Reinsberg, H. Baltruschat, Inverse modelling of thin layer flow cells and RRDEs, The 67th Annual Meeting of the International Society of Electrochemistry, Den Haag, Netherlands, August 21  26, 2016.

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

P.É. Druet, Existence of global weak solutions for generalized PoissonNernstPlanck systems, 7th European Congress of Mathematics (ECM), minisymposium ``Analysis of Thermodynamically Consistent Models of Electrolytes in the Context of Battery Research'', July 18  22, 2016, Technische Universität Berlin, Berlin, July 20, 2016.

J. Fuhrmann, Ch. Merdon, A thermodynamically consistent numerical approach to NernstPlanckPoisson 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.

J. Fuhrmann, A. Linke, Ch. Merdon, M. Khodayari , H. Baltruschat, Detection of solubility, transport and reaction coefficients from experimental data by inverse modelling of thin layer flow cells, 1st Leibniz MMS Mini Workshop on CFD & GFD, WIAS Berlin, September 8  9, 2016.

J. Fuhrmann, A. Linke, Ch. Merdon, W. Dreyer, C. Guhle, M. Landstorfer, R. Müller, Numerical methods for electrochemical systems, 2nd Graz Battery Days, Graz, Austria, September 27  28, 2016.

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

CH. Merdon, Inverse modeling of thin layer flow cells for detection of solubility transport and reaction coefficients from experimental data, 17th Topical Meeting of the International Society of Electrochemistry Multiscale Analysis of Electrochemical Systems, May 31  June 3, 2015, Saint Malo Congress Center, France, June 1, 2015.

A. Mielke, Chemical Master Equation: Coarse graining via gradient structures, Kolloquium des SFB 1114 ``Scaling Cascades in Complex Systems'', Freie Universität Berlin, Fachbereich Mathematik, Berlin, June 4, 2015.

A. Mielke, Evolutionary $Gamma$convergence for generalized gradient systems, Workshop ``Gradient Flows'', June 22  23, 2015, Université Pierre et Marie Curie, Laboratoire JacquesLouis Lions, Paris, France, June 22, 2015.

M. Liero, On dissipation distances for reactiondiffusion equations  The HellingerKantorovich distance, Workshop ``Entropy Methods, PDEs, Functional Inequalities, and Applications'', June 30  July 4, 2014, Banff International Research Station for Mathematical Innovation and Discovery (BIRS), Canada, July 1, 2014.

M. Liero, On dissipation distances for reactiondiffusion equations  The HellingerKantorovich distance, RIPE60  Rate Independent Processes and Evolution Workshop, June 24  26, 2014, Prague, Czech Republic, June 24, 2014.

A. Linke, Ch. Merdon, Optimal and pressureindependent $L^2$ velocity error estimates for a modified CrouzeixRaviart element with BDM reconstructions, The International Symposium of Finite Volumes for Complex Applications VII (FVCA 7), BerlinBrandenburgische Akademie der Wissenschaften, June 15  20, 2014.

A. Fiebach, A. Glitzky, Uniform estimate of the relative free energy by the dissipation rate for finite volume discretized reactiondiffusion systems, The International Symposium of Finite Volumes for Complex Applications VII (FVCA 7), Berlin, June 15  20, 2014.

A. Glitzky, Driftdiffusion models for heterostructures in photovoltaics, 8th European Conference on Elliptic and Parabolic Problems, Minisymposium ``Qualitative Properties of Nonlinear Elliptic and Parabolic Equations'', May 26  30, 2014, Universität Zürich, Institut für Mathematik, organized in Gaeta, Italy, May 27, 2014.

M. Thomas, Thermomechanical modeling of dissipative processes in elastic media via energy and entropy, The 10th AIMS Conference on Dynamical Systems, Differential Equations and Applications, Special Session 8: Emergence and Dynamics of Patterns in Nonlinear Partial Differential Equations from Mathematical Science, July 7  11, 2014, Madrid, Spain, July 8, 2014.

J. Fuhrmann, A. Linke, Ch. Merdon, M. Khodayari, H. Baltruschat, Detection of solubility, transport and reaction coefficients from experimental data by inverse modeling of thin layer flow cells, 65th Annual Meeting of the International Society of Electrochemistry, Lausanne, Switzerland, August 31  September 5, 2014.

J. Fuhrmann, A. Linke, Ch. Merdon, Coupling of fluid flow and solute transport using a divergencefree reconstruction of the CrouzeixRaviart element, The International Symposium of Finite Volumes for Complex Applications VII (FVCA 7), BerlinBrandenburgische Akademie der Wissenschaften, June 15  20, 2014.

J. Fuhrmann, Activity based finite volume methods for generalised NernstPlanckPoisson systems, The International Symposium of Finite Volumes for Complex Applications VII (FVCA 7), BerlinBrandenburgische Akademie der Wissenschaften, June 15  20, 2014.

A. Mielke, On a metric and geometric approach to reactiondiffusion systems as gradient systems, Mathematics Colloquium, Jacobs University Bremen, School of Engineering and Science, December 1, 2014.

A. Mielke, A reactiondiffusion equation as a HellingerKantorovich gradient flow, ERC Workshop on Optimal Transportation and Applications, October 27  31, 2014, Centro di Ricerca Matematica ``Ennio De Giorgi'', Pisa, Italy, October 29, 2014.

J. Rehberg, On nonsmooth parabolic equations, Workshop ``MaxwellStefan meets NavierStokes/Modeling and Analysis of Reactive MultiComponent Flows'', March 31  April 2, 2014, Universität Halle, April 1, 2014.

J. Rehberg, Optimal Sobolev regularity for second order divergence operators, 85th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2014), Session on Applied Operator Theory, March 10  14, 2014, FriedrichAlexander Universität ErlangenNürnberg, March 13, 2014.

S. Reichelt, Homogenization of degenerated reactiondiffusion equations, Doktorandenforum der LeibnizGemeinschaft, Sektion D, Berlin, June 6  7, 2013.

M. Liero, On gradient structures for driftreactiondiffusion systems and Markov chains, Analysis Seminar, University of Bath, Mathematical Sciences, UK, November 21, 2013.

M. Liero, Gradient structures and geodesic convexity for reactiondiffusion system, SIAM Conference on Mathematical Aspects of Materials Science (MS13), Minisymposium ``Material Modelling and Gradient Flows'' (MS100), June 9  12, 2013, Philadelphia, USA, June 12, 2013.

M. Liero, On gradient structures and geodesic convexity for reactiondiffusion systems, Research Seminar, Westfälische WilhelmsUniversität Münster, Institut für Numerische und Angewandte Mathematik, April 17, 2013.

A. Glitzky, Continuous and finite volume discretized reactiondiffusion systems in heterostructures, Asymptotic Behaviour of Systems of PDE Arising in Physics and Biology: Theoretical and Numerical Points of View, November 6  8, 2013, Lille 1 University  Science and Technology, France, November 6, 2013.

A. Mielke, Gradient structures and uniform global decay for reactiondiffusion systems, Mathematisches Kolloquium, Universität Bielefeld, Fakultät für Mathematik, April 25, 2013.

A. Mielke, On the geometry of reactiondiffusion systems: Optimal transport versus reaction, Recent Trends in Differential Equations: Analysis and Discretisation Methods, November 7  9, 2013, Technische Universität Berlin, Institut für Mathematik, November 9, 2013.

A. Mielke, Using gradient structures for modeling semiconductors, Eindhoven University of Technology, Institute for Complex Molecular Systems, Netherlands, February 21, 2013.

M. Liero, Interfaces in reactiondiffusion systems, Seminar ``Dünne Schichten'', Technische Universität Berlin, Institut für Mathematik, February 9, 2012.

A. Glitzky, An electronic model for solar cells taking into account active interfaces, International Workshop ``Mathematics for Semiconductur Heterostructures: Modeling, Analysis, and Numerics'', September 24  28, 2012, WIAS Berlin, September 27, 2012.

M. Thomas, Thermomechanical modeling via energy and entropy, Seminar on Applied Mathematics, University of Pavia, Department of Mathematics, Italy, February 14, 2012.

M. Thomas, Thermomechanical modeling via energy and entropy using GENERIC, Workshop ``Mechanics of Materials'', March 19  23, 2012, Mathematisches Forschungsinstitut Oberwolfach, March 22, 2012.

A. Mielke, Entropy gradient flows for Markow chains and reactiondiffusion systems, BerlinLeipzigSeminar ``Analysis/Probability Theory'', WIAS Berlin, April 13, 2012.

A. Mielke, GradientenStrukturen und geodätische Konvexität für MarkovKetten und ReaktionsDiffusionsSysteme, Augsburger Kolloquium, Universität Augsburg, Institut für Mathematik, May 8, 2012.

A. Mielke, Multidimensional modeling and simulation of optoelectronic devices, Challenge Workshop ``Modeling, Simulation and Optimisation Tools'', September 24  26, 2012, Technische Universität Berlin, September 24, 2012.

A. Mielke, On geodesic convexity for reactiondiffusion systems, Seminar on Applied Mathematics, Università di Pavia, Dipartimento di Matematica, Italy, March 6, 2012.

A. Mielke, On gradient flows and reactiondiffusion systems, Institutskolloquium, MaxPlanckInstitut für Mathematik in den Naturwissenschaften, Leipzig, December 3, 2012.

A. Mielke, On gradient structures and geodesic convexity for energyreactiondiffusion systems and Markov chains, ERC Workshop on Optimal Transportation and Applications, November 5  9, 2012, Centro di Ricerca Matematica ``Ennio De Giorgi'', Pisa, Italy, November 8, 2012.

A. Mielke, On gradient structures for Markov chains and reactiondiffusion systems, Applied & Computational Analysis (ACA) Seminar, University of Cambridge, Department of Applied Mathematics and Theoretical Physics (DAMTP), UK, June 14, 2012.

A. Mielke, Using gradient structures for modeling semiconductors, International Workshop ``Mathematics for Semiconductur Heterostructures: Modeling, Analysis, and Numerics'', September 24  28, 2012, WIAS Berlin, September 24, 2012.

A. Mielke, Thermodynamical modeling of bulkinterface interaction in reactiondiffusion systems, Interfaces and Discontinuities in Solids, Liquids and Crystals (INDI2011), June 20  23, 2011, Gargnano (Brescia), Italy, June 20, 2011.

A. Mielke, Mathematical approaches to thermodynamic modeling, Autumn School on Mathematical Principles for and Advances in Continuum Mechanics, November 7  12, 2011, Centro di Ricerca Matematica ``Ennio De Giorgi'', Pisa, Italy.

A. Glitzky, Uniform exponential decay of the free energy for Voronoi finite volume discretized reactiondiffusion systems, 8th AIMS International Conference on Dynamical Systems, Differential Equations and Applications, Special Session on Reaction Diffusion Systems, May 25  28, 2010, Technische Universität Dresden, May 26, 2010.

A. Mielke, Gradient structures for reactiondiffusion systems and semiconductor equations, 81th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM 2010), Session on Applied Analysis, March 22  26, 2010, Universität Karlsruhe, March 24, 2010.

A. Linke, Divergencefree mixed finite elements for the incompressible NavierStokes equations, Universität Stuttgart, Institut für Wasserbau, December 8, 2009.

M. Ehrhardt, J. Fuhrmann, A. Linke, Finite volume methods for the simulation of flow cell experiments, Workshop ``New Trends in Model Coupling  Theory, Numerics & Applications'' (NTMC'09), Paris, France, September 2  4, 2009.

M. Ehrhardt, The fluidporous interface problem: Analytic and numerical solutions to flow cell problems, 6th Symposium on Fuel Cell Modelling and Experimental Validation (MODVAL 6), March 25  26, 2009, Evangelische Akademie Baden, Bad Herrenalb, March 26, 2009.

M. Ehrhardt, The fluidporous interface problem: Analytic and numerical solutions to flow cell problems, Mathematical Models in Medicine, Business, Engineering (XI JORNADAS IMM), September 8  11, 2009, Technical University of Valencia, Institute of Multidisciplinary Mathematics, Spain, September 10, 2009.

J. Fuhrmann, Mathematical and numerical models of electrochemical processes related to porous media, International Conference on Nonlinearities and Upscaling in Porous Media (NUPUS), October 5  7, 2009, Universität Stuttgart, October 6, 2009.

J. Fuhrmann, Model based numerical impedance calculation in electrochemical systems, 6th Symposium on Fuel Cell Modelling and Experimental Validation (MODVAL 6), March 25  26, 2009, Evangelische Akademie Baden, Bad Herrenalb, March 25, 2009.

J. Fuhrmann, Numerical modeling in electrochemistry, Conference on Scientific Computing (ALGORITMY 2009), March 15  20, 2009, Slovak University of Technology, Department of Mathematics and Descriptive Geometry, Podbanské, March 17, 2005.

A. Linke, Mass conservative coupling of fluid flow and species transport in electrochemical flow cells, 13th Conference on Mathematics of Finite Elements and Applications (MAFELAP 2009), June 9  12, 2009, Brunel University, London, UK, June 10, 2009.

A. Linke, The discretization of coupled flows and the problem of mass conservation, Workshop on Discretization Methods for Viscous Flows, Part II: Compressible and Incompressible Flows, June 24  26, 2009, Porquerolles, Toulon, France, June 25, 2009.

A. Linke, The discretization of coupled flows and the problem of mass conservation, Seventh Negev Applied Mathematical Workshop, July 6  8, 2009, Ben Gurion University of the Negev, Jacob Blaustein Institute for Desert Research, Sede Boqer Campus, Israel, July 7, 2009.

J. Rehberg, Quasilinear parabolic equations in distribution spaces, International Conference on Nonlinear Parabolic Problems in Honor of Herbert Amann, May 10  16, 2009, Stefan Banach International Mathematical Center, Bedlewo, Poland, May 12, 2009.

E. Holzbecher, H. Zhao, J. Fuhrmann, A. Linke, H. Langmach, Numerical investigation of thin layer flow cells, 4th Gerischer Symposium, Berlin, June 25  27, 2008.

E. Bänsch, H. Berninger, U. Böhm, A. Bronstert, M. Ehrhardt, R. Forster, J. Fuhrmann, R. Klein, R. Kornhuber, A. Linke, A. Owinoh, J. Volkholz, Pakt für Forschung und Innovation: Das Forschungsnetzwerk ``Gekoppelte Strömungsprozesse in Energie und Umweltforschung'', Show of the Leibniz Association ``Exzellenz durch Vernetzung. Kooperationsprojekte der deutschen Wissenschaftsorganisationen mit Hochschulen im Pakt für Forschung und Innovation'', Berlin, November 12, 2008.

M. Ehrhardt, O. Gloger, Th. Dietrich, O. Hellwich, K. Graf, E. Nagel, Level Set Methoden zur Segmentierung von kardiologischen MRBildern, 22. Treffpunkt Medizintechnik: Fortschritte in der medizinischen Bildgebung, Charité, Campus Virchow Klinikum Berlin, May 22, 2008.

A. Glitzky, Energy estimates for continuous and discretized reactiondiffusion systems in heterostructures, Annual Meeting of the Deutsche MathematikerVereinigung 2008, minisymposium ``Analysis of ReactionDiffusion Systems with Internal Interfaces'', September 15  19, 2008, FriedrichAlexanderUniversität ErlangenNürnberg, September 15, 2008.

A. Glitzky, Energy estimates for space and time discretized electroreactiondiffusion systems, Conference on Differential Equations and Applications to Mathematical Biology, June 23  27, 2008, Université Le Havre, France, June 26, 2008.

A. Linke, Mass conservative coupling of fluid flow and species transport in electrochemical flow cells, Annual Meeting of the Deutsche MathematikerVereinigung 2008, September 15  19, 2008, FriedrichAlexanderUniversität ErlangenNürnberg, September 16, 2008.

A. Linke, Mass conservative coupling of fluid flow and species transport in electrochemical flow cells, GeorgAugustUniversität Göttingen, November 11, 2008.

J. Rehberg, Hölder continuity for elliptic and parabolic problems, AnalysisTag, Technische Universität Darmstadt, Fachbereich Mathematik, November 27, 2008.

A. Glitzky, Energy estimates for reactiondiffusion processes of charged species, 6th International Congress on Industrial and Applied Mathematics (ICIAM), July 16  20, 2007, ETH Zürich, Switzerland, July 16, 2007.

J. Rehberg, An elliptic model problem including mixed boundary conditions and material heterogeneities, Fifth Singular Days, April 23  27, 2007, International Center for Mathematical Meetings, Luminy, France, April 26, 2007.

J. Rehberg, Maximal parabolic regularity on Sobolev spaces, The Eighteenth Crimean Autumn Mathematical SchoolSymposium (KROMSH2007), September 17  29, 2007, LaspiBatiliman, Ukraine, September 18, 2007.

F. Schmid, An evolution model in contact mechanics with dry friction, 6th International Congress on Industrial and Applied Mathematics (ICIAM), July 16  20, 2007, ETH Zürich, Switzerland, July 19, 2007.

A. Glitzky, Energy estimates for electroreactiondiffusion systems with partly fast kinetics, 6th AIMS International Conference on Dynamical Systems, Differential Equations & Applications, June 25  28, 2006, Université de Poitiers, France, June 27, 2006.

J. Rehberg, Existence and uniqueness for van Roosbroeck's system in Lebesque spaces, Conference ``Recent Advances in Nonlinear Partial Differential Equations and Applications'', Toledo, Spain, June 7  10, 2006.

J. Rehberg, Regularity for nonsmooth elliptic problems, Crimean Autumn Mathematical School, September 20  25, 2006, Vernadskiy Tavricheskiy National University, Laspi, Ukraine, September 21, 2006.

J. Rehberg, Elliptische und parabolische Probleme aus Anwendungen, Kolloquium im Fachbereich Mathematik, Universität Darmstadt, May 18, 2005.

J. Rehberg, Existence, uniqeness and regularity for quasilinear parabolic systems, International Conference ``Nonlinear Partial Differential Equations'', September 17  24, 2005, Institute of Applied Mathematics and Mechanics Donetsk, Alushta, Ukraine, September 18, 2005.

J. Rehberg, H$^1,q$regularity for linear, elliptic boundary value problems, Regularity for nonlinear and linear PDEs in nonsmooth domains  Analysis, simulation and application, September 5  7, 2005, Universität Stuttgart, Deutsche Forschungsgemeinschaft (SFB 404), Hirschegg, Austria, September 6, 2005.

J. Rehberg, Regularität für elliptische Probleme mit unglatten Daten, Oberseminar Prof. Escher/Prof. Schrohe, Technische Universität Hannover, December 13, 2005.

J. Rehberg, Analysis of macroscopic and quantum mechanical semiconductor models, International Visitor Program ``Nonlinear Parabolic Problems'', August 8  November 18, 2005, Finnish Mathematical Society (FMS), University of Helsinki, and Helsinki University of Technology, Finland, November 1, 2005.

J. Rehberg, Existence, uniqueness and regularity for quasilinear parabolic systems, Conference ``Nonlinear Parabolic Problems'', October 17  21, 2005, Finnish Mathematical Society (FMS), University of Helsinki, and Helsinki University of Technology, Finland, October 20, 2005.

J. Rehberg, Elliptische und parabolische Probleme mit unglatten Daten, Technische Universität Darmstadt, Fachbereich Mathematik, December 14, 2004.

J. Rehberg, Quasilinear parabolic equations in $L^p$, Nonlinear Elliptic and Parabolic Problems: A Special Tribute to the Work of Herbert Amann, June 28  30, 2004, Universität Zürich, Institut für Mathematik, Switzerland, June 29, 2004.

J. Rehberg, The twodimensional van Roosbroeck system has solutions in $L^p$, Workshop ``Advances in Mathematical Semiconductor Modelling: Devices and Circuits'', March 2  6, 2004, ChineseGerman Centre for Science Promotion, Beijing, China, March 5, 2004.

A. Glitzky, R. Hünlich, Stationary solutions of twodimensional heterogeneous energy models with multiple species, Nonlocal Elliptic and Parabolic Problems, September 9  11, 2003, Bedlewo, Poland, September 10, 2003.

H.Chr. Kaiser, On space discretization of reactiondiffusion systems with discontinuous coefficients and mixed boundary conditions, 2nd GAMM Seminar on Microstructures, January 10  11, 2003, RuhrUniversität Bochum, Institut für Mechanik, January 10, 2003.

J. Rehberg, Solvability and regularity for parabolic equations with nonsmooth data, International Conference ``Nonlinear Partial Differential Equations'', September 15  21, 2003, National Academy of Sciences of Ukraine, Institute of Applied Mathematics and Mechanics, Alushta, September 17, 2003.
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Projects/Grants
 Investigation of Electrochemical Double Layers in Solid Oxide Cells (EDLSOC)
 Modellbasierte Abschätzung der Lebensdauer von gealterten LiBatterien für die 2ndLife Anwendung als stationärer Stromspeicher (MALLi2)
 MultiMaterial Electrocatalysis (MULTECAT)
 Sensing with Nanopores
 Verbundprojekt “Wege zu sekundären Mg/Ca  LuftBatterien”