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- 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.
- ItemA higher gradient theory of mixtures for multi-component materials with numerical examples for binary alloys(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2007) Böhme, Thomas; Dreyer, Wolfgang; Duderstadt, Frank; Müller, Wolfgang H.A theory of mixture for multi-component materials is presented based on a novel, straightforward method for the exploitation of the Second Law of thermodynamics. In particular the constitutive equations for entropy, heat and diffusion flux as well as the stress tensor are formulated as a consequence of the non-negative entropy production. Furthermore we derive the established Gibbs equation as well as the Gibbs Duhem relation which also follow from the formalism. Moreover, it is illustrated, how local mechanical strains due to eigenstrains or external loadings, modify the free energy and, consequently, change the chemical potentials of the components. All consecutive steps are illustrated, first, for simple mixtures and, second, for a system containing two different phases. So-called higher gradients of the concentrations are considered, which take the nonuniform composition into account. It will also become apparent that more/other variables of modified/different physical pr oblems beyond the illustrated ones can be easily treated within the presented framework. This work ends with the specification to binary alloys and with the presentation of various numerical simulations.
- ItemMulticomponent incompressible fluids -- An asymptotic study(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2021) Bothe, Dieter; Dreyer, Wolfgang; Druet, Pierre-ÉtienneThis paper investigates the asymptotic behavior of the Helmholtz free energy of mixtures at small compressibility. We start from a general representation for the local free energy that is valid in stable subregions of the phase diagram. On the basis of this representation we classify the admissible data to construct a thermodynamically consistent constitutive model. We then analyze the incompressible limit, where the molar volume becomes independent of pressure. Here we are confronted with two problems: (i) Our study shows that the physical system at hand cannot remain incompressible for arbitrary large deviations from a reference pressure unless its volume is linear in the composition. (ii) As a consequence of the 2nd law of thermodynamics, the incompressible limit implies that the molar volume becomes independent of temperature as well. Most applications, however, reveal the non-appropriateness of this property. According to our mathematical treatment, the free energy as a function of temperature and partial masses tends to a limit in the sense of epi-- or Gamma--convergence. In the context of the first problem, we study the mixing of two fluids to compare the linearity with experimental observations. The second problem will be treated by considering the asymptotic behavior of both a general inequality relating thermal expansion and compressibility and a PDE-system relying on the equations of balance for partial masses, momentum and the internal energy.
- ItemHysteresis in the context of hydrogen storage and lithium-ion batteries(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2009) Dreyer, Wolfgang; Guhlke, Clemens; Huth, RobertThe processes of reversible storage of hydrogen in a metal by loading and unloading and of charging and discharging of lithium-ion batteries have many things in common. The both processes are accompanied by a phase transition and loading and unloading run along different paths, so that hysteretic behavior is observed. For hydrogen storage we consider a fine powder of magnesium (Mg) particles and lithium storage is studied for iron phosphate (FePO_4) particles forming the cathode of a lithium-ion battery. The mathematical models that are established in citeDGJ08 and citeDGH09a, describe phase transitions and hysteresis exclusively in a single particle and on that basis they can predict the observed hysteretic plots with almost horizontal plateaus. Interestingly the models predict that the coexistence of a 2-phase system in an individual particle disappears, if its size is below a critical value. However, measurements reveal that this is qualitatively not reflected by the mentioned hysteretic plots of loading and unloading. In other words: The behavior of a storage system consisting of many particles is qualitatively independent of the fact whether the individual particles itself develop a 2-phase system or if they remain in a single phase state. This apparent paradoxical observation will be resolved in this article. It will be shown that if each of the individual particles homogeneously distributes the supplied matter, nevertheless the many particle ensemble exhibits phase transition and hysteresis, because one of the two phases is realized in some part of the particles while the remaining part is in the other phase.
- ItemNew insights on the interfacial tension of electrochemical interfaces and the Lippmann equation(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2015) Dreyer, Wolfgang; Guhlke, Clemens; Landstorfer, Manuel; Neumann, Johannes; Müller, RüdigerThe 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 radius is comparable to the Debye length.
- ItemModelling compressible electrolytes with phase transition(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2014) Dreyer, Wolfgang; Giesselmann, Jan; Kraus, ChristianeA novel thermodynamically consistent diffuse interface model is derived for compressible electrolytes with phase transitions. The fluid mixtures may consist of N constituents with the phases liquid and vapor, where both phases may coexist. In addition, all constituents may consist of polarizable and magnetizable matter. Our introduced thermodynamically consistent diffuse interface model may be regarded as a generalized model of Allen-Cahn/Navier-Stokes/Poisson type for multi-component flows with phase transitions and electrochemical reactions. For the introduced diffuse interface model, we investigate physically admissible sharp interface limits by matched asymptotic techniques. We consider two scaling regimes, i.e. a non-coupled and a coupled regime, where the coupling takes place between the smallness parameter in the Poisson equation and the width of the interface. We recover in the sharp interface limit a generalized Allen-Cahn/Euler/Poisson system for mixtures with electrochemical reactions in the bulk phases equipped with admissible interfacial conditions. The interfacial conditions satisfy, for instance, a generalized Gibbs-Thomson law and a dynamic Young-Laplace law.
- ItemPhase transition and hysteresis in a rechargeable lithium battery(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2007) Dreyer, Wolfgang; Gaberšček, Miran; Jamnik, JankoWe represent a model which describes the evolution of a phase transition that occurs in some part of a rechargeable lithium battery during the process of charging/discharging. The model is capable to simulate the hysteretic behavior of the voltage - charge characteristics. During discharging of the battery, the interstitial lattice sites of a small crystalline host system are filled up with lithium atoms and these are released again during charging. We show within the context of a sharp interface model that two mechanical phenomena go along with a phase transition that appears in the host system during supply and removal of lithium. At first the lithium atoms need more space than it is available by the interstitial lattice sites, which leads to a maximal relative change of the crystal volume of about $6%$. Furthermore there is an interface between two adjacent phases that has very large curvature of the order of magnitude 100 m, which evoke here a discontinuity of the normal component of the stress. In order to simulate the dynamics of the phase transitions and in particular the observed hysteresis we establish a new initial and boundary value problem for a nonlinear PDE system that can be reduced in some limiting case to an ODE system.
- ItemA diffuse interface model for quasi-incrompressible flows : sharp interface limits and numerics(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2012) Aki, Gonca; Daube, Johannes; Dreyer, Wolfgang; Giesselmann, Jan; Kränkel, Mirko; Kraus, ChristianeIn this contribution, we investigate a diffuse interface model for quasi–incompressible flows. We determine corresponding sharp interface limits of two different scalings. The sharp interface limit is deduced by matched asymptotic expansions of the fields in powers of the interface. In particular, we study solutions of the derived system of inner equations and discuss the results within the general setting of jump conditions for sharp interface models. Furthermore, we treat, as a subproblem, the convective Cahn–Hilliard equation numerically by a Local Discontinuous Galerkin scheme.
- ItemHysteresis and phase transition in many-particle storage systems(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2010) Dreyer, Wolfgang; Guhlke, Clemens; Herrmann, MichaelWe study the behavior of systems consisting of ensembles of interconnected storage particles. Our examples concern the storage of lithium in many-particle electrodes of rechargeable lithium-ion batteries and the storage of air in a system of interconnected rubber balloons. We are particularly interested in those storage systems whose constituents exhibit non-monotone material behavior leading to transitions between two coexisting phases and to hysteresis. In the current study we consider the case that the time to approach equilibrium of a single storage particle is much smaller than the time for full charging of the ensemble. In this regime the evolution of the probability to find a particle of the ensemble in a certain state, may be described by a nonlocal conservation law of Fokker-Planck type. Two constant parameter control whether the ensemble transits the 2-phase region along a Maxwell line or along a hysteresis path or if the ensemble shows the same non-monotone behavior as its constituents.
- ItemExistence of weak solutions for improved Nernst-Planck-Poisson models of compressible reacting electrolytes(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2016) Dreyer, Wolfgang; Druet, Pierre-Étienne; Gajewski, Paul; Guhlke, ClemensWe consider an improved Nernst-Planck-Poisson model for compressible electrolytes first proposed by Dreyer et al. in 2013. The model takes into account the elastic deformation of the medium. In particular, large pressure contributions near electrochemical interfaces induce 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 the Poisson equation for the electrical potential. Cross-diffusion phenomena occur due to the principle of mass conservation. Moreover, the diffusion matrix (mobility matrix) has a zero eigenvalue, meaning that the system is degenerate parabolic. In this paper we establish the existence of a global-in-time weak solution for the full model, allowing for cross-diffusion and an arbitrary number of chemical reactions in the bulk and on the active boundary.