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- ItemA continuum model for yttria-stabilized zirconia incorporating triple phase boundary, lattice structure and immobile oxide ions(Berlin ; Heidelberg ; New York : Springer, 2019) Vágner, Petr; Guhlke, Clemens; Miloš, Vojtěch; Müller, Rüdiger; Fuhrmann, JürgenA continuum model for yttria-stabilized zirconia (YSZ) in the framework of non-equilibrium thermodynamics is developed. Particular attention is given to (i) modeling of the YSZ-metal-gas triple phase boundary, (ii) incorporation of the lattice structure and immobile oxide ions within the free energy model and (iii) surface reactions. A finite volume discretization method based on modified Scharfetter-Gummel fluxes is derived in order to perform numerical simulations. The model is used to study the impact of yttria and immobile oxide ions on the structure of the charged boundary layer and the double layer capacitance. Cyclic voltammograms of an air-half cell are simulated to study the effect of parameter variations on surface reactions, adsorption and anion diffusion. © 2019, The Author(s).
- ItemBulk-Surface Electrothermodynamics and Applications to Electrochemistry(Basel : MDPI, 2018) Dreyer, Wolfgang; Guhlke, Clemens; Müller, RüdigerWe 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 of 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.
- ItemOptimal and robust a posteriori error estimates in L∞(L2) for the approximation of Allen-Cahn equations past singularities(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2009) Bartels, Sören; Müller, RüdigerOptimal a posteriori error estimates in L∞(L2) are derived for the finite element approximation of Allen-Cahn equations. The estimates depend on the inverse of a small parameter only in a low order polynomial and are valid past topological changes of the evolving interface. The error analysis employs an elliptic reconstruction of the approximate solution and applies to a large class of conforming, nonconforming, mixed, and discontinuous Galerkin methods. Numerical experiments illustrate the theoretical results.
- ItemError control for the approximation of Allen-Cahn and Cahn-Hilliard equations with a logarithmic potential(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2010) Bartels, Sören; Müller, RüdigerA fully computable upper bound for the finite element approximation error of Allen-Cahn and Cahn-Hilliard equations with logarithmic potentials is derived. Numerical experiments show that for the sharp interface limit this bound is robust past topological changes. Modifications of the abstract results to derive quasi-optimal error estimates in different norms for lowest order finite element methods are discussed and lead to weaker conditions on the residuals under which the conditional error estimates hold.
- ItemOvercoming the shortcomings of the Nernst-Planck model(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2012) Dreyer, Wolfgang; Guhlke, Clemens; Müller, RüdigerThis 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 Nernst-Planck model. A boundary problem for equilibrium processes is used to illustrate the new features of our model.
- ItemStable Crank-Nicolson discretisation for incompressible miscible displacement problems of low regularity(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2009) Jensen, Max; Müller, RüdigerIn this article we study the numerical approximation of incompressible miscible displacement problems with a linearised Crank-Nicolson time discretisation, combined with a mixed finite element and discontinuous Galerkin method. At the heart of the analysis is the proof of convergence under low regularity requirements. Numerical experiments demonstrate that the proposed method exhibits second-order convergence for smooth and robustness for rough problems.
- ItemA new perspective on the electron transfer: Recovering the Butler-Volmer equation in non-equilibrium thermodynamics(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Dreyer, Wolfgang; Guhlke, Clemens; Müller, RüdigerUnderstanding and correct mathematical description of electron transfer reaction is a central question in electrochemistry. Typically the electron transfer reactions are described by the Butler-Volmer equation which has its origin in kinetic theories. The Butler-Volmer equation relates interfacial reaction rates to bulk quantities like the electrostatic potential and electrolyte concentrations. Since in the classical form, the validity of the Butler-Volmer equation is limited to some simple electrochemical systems, many attempts have been made to generalize the Butler-Volmer equation. Based on non-equilibrium thermodynamics we have recently derived a reduced model for the electrode-electrolyte interface. This reduced model includes surface reactions and adsorption but does not resolve the charge layer at the interface. Instead it is locally electroneutral and consistently incorporates all features of the double layer into a set of interface conditions. In the context of this reduced model we are able to derive a general Butler-Volmer equation. We discuss the application of the new Butler-Volmer equations to different scenarios like electron transfer reactions at metal electrodes, the intercalation process in lithium-iron-phosphate electrodes and adsorption processes. We illustrate the theory by an example of electroplating.
- ItemModeling polycrystalline electrode-electrolyte interfaces: The differential capacitance(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2019) Fuhrmann, Jürgen; Landstorfer, Manuel; Müller, RüdigerWe present and analyze a model for polycrystalline electrode surfaces based on an improved continuum model that takes finite ion size and solvation into account. The numerical simulation of finite size facet patterns allows to study two limiting cases: While for facet size diameter $d^facet to 0$ we get the typical capacitance of a spatially homogeneous but possible amorphous or liquid surface, in the limit $L^Debye << d^facet$ , an ensemble of non-interacting single crystal surfaces is approached. Already for moderate size of the facet diameters, the capacitance is remarkably well approximated by the classical approach of adding the single crystal capacities of the contributing facets weighted by their respective surface fraction. As a consequence, the potential of zero charge is not necessarily attained at a local minimum of capacitance, but might be located at a local capacitance maximum instead. Moreover, the results show that surface roughness can be accurately taken into account by multiplication of the ideally flat polycrystalline surface capacitance with a single factor. In particular, we find that the influence of the actual geometry of the facet pattern in negligible and our theory opens the way to a stochastic description of complex real polycrystal surfaces.
- ItemThe impact of solvation and dissociation on the transport parameters of liquid electrolytes: Continuum modeling and numerical study(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2018) Dreyer, Wolfgang; Guhlke, Clemens; Müller, RüdigerElectro-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 MaxwellStefan 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.
- ItemRational modeling of electrochemical double-layers and derivation of Butler-Volmer equations(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) Dreyer, Wolfgang; Guhlke, Clemens; Müller, RüdigerWe derive the boundary conditions for the contact between an electrolyte and a solid electrode. At first we revisit the thermodynamic consistent complete model that resolves the actual electrodeelectrolyte interface and its adjacent boundary layers. The width of these layers is controlled by the Debye length that is typically very small, leading to strongly different length scales in the system. We 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. This approach fully determines the relation of bulk quantities to the boundary conditions of the reduced model. In particular, the Butler-Volmer equations for electrochemical reactions, which are still under discussion in the literature, are rational consequences of our approach. For illustration and to compare with the literature, we consider a simple generic reaction.