Browsing by Author "Dreyer, Wolfgang"
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- ItemAnalysis of improved Nernst-Planck-Poisson models of compressible isothermal electrolytes. Part I: Derivation of the model and survey of the results(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2017) Dreyer, Wolfgang; Druet, Pierre-Étienne; Gajewski, Paul; Guhlke, ClemensWe 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 Navier-Stokes 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.
- ItemAnalysis of improved Nernst-Planck-Poisson models of compressible isothermal electrolytes. Part II: Approximation and a priori estimates(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2017) Dreyer, Wolfgang; Druet, Pierre-Étienne; Gajewski, Paul; Guhlke, ClemensWe 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 Navier-Stokes 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.
- ItemAnalysis of improved Nernst-Planck-Poisson models of compressible isothermal electrolytes. Part III: Compactness and convergence(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2017) Dreyer, Wolfgang; Druet, Pierre-Étienne; Gajewski, Paul; Guhlke, ClemensWe consider an improved Nernst-Planck-Poisson model first proposed by Dreyer et al. in 2013 for compressible isothermal electrolytes in non equilibrium. The model takes into account the elastic deformation of the medium that induces an inherent coupling of mass and momentum transport. The model consists of convectiondiffusionreaction equations for the constituents of the mixture, of the Navier-Stokes equation for the barycentric velocity, and of the Poisson equation for the electrical potential. Due to the principle of mass conservation, 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.
- ItemAnalysis of improved Nernst–Planck–Poisson models of compressible isothermal electrolytes(Cham (ZG) : Springer International Publishing AG, 2020) Dreyer, Wolfgang; Druet, Pierre-Étienne; Gajewski, Paul; Guhlke, ClemensWe consider an improved Nernst–Planck–Poisson model first proposed by Dreyer et al. in 2013 for compressible isothermal electrolytes in non-equilibrium. The elastic deformation of the medium, that induces an inherent coupling of mass and momentum transport, is taken into account. The model consists of convection–diffusion–reaction equations for the constituents of the mixture, of the Navier–Stokes equation for the barycentric velocity and of the Poisson equation for the electrical potential. Due to the principle of mass conservation, cross-diffusion phenomena must occur, and the mobility matrix (Onsager matrix) has a non-trivial kernel. In this paper, we establish the existence of a global-in-time weak solution, allowing for a general structure of the mobility tensor and for chemical reactions with fast nonlinear 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.
- ItemAsymptotic analysis for Korteweg models(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2010) Dreyer, Wolfgang; Giesselmann, Jan; Kraus, Christiane; Rohde, ChristianeThis paper deals with a sharp interface limit of the isothermal Navier-Stokes-Korteweg system. The sharp interface limit is performed by matched asymptotic expansions of the fields in powers of the interface width. These expansions are considered in the interfacial region (inner expansions) and in the bulk (outer expansion) and are matched order by order. Particularly we consider the first orders of the corresponding inner equations obtained by a change of coordinates in an interfacial layer. For a specific scaling we establish solvability criteria for these inner equations and recover the results within the general setting of jump conditions for sharp interface models.
- ItemThe behavior of a many particle cathode in a lithium-ion battery(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2009) Dreyer, Wolfgang; Guhlke, Clemens; Huth, RobertWe study the almost reversible storage process of charging and discharging of lithium-ion batteries. That process is accompanied by a phase transition and charging and discharging run along different paths, so that hysteretic behavior is observed. We are interested in the storage problem of the cathode of a lithium-ion battery consisting of a system of many iron phosphate (FePO4) particles. There are mathematical models, see [DGJ08], [DGGHJ09] and [DG09], that describe phase transitions and hysteresis exclusively in a single storage particle and they can describe the observed hysteretic voltage-charge 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. The disappearance of the phase transition in the single particle model implies the disappearance of the hysteresis. However, in the experiment hysteretic behavior survives. 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 by a many particle model. It will be shown that if each of the individual particles is in a homogeneous state, nevertheless the many particle ensemble exhibits phase transition and hysteresis ...
- ItemBlow-up versus boundedness in a nonlocal and nonlinear Fokker-Planck equation(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2011) Dreyer, Wolfgang; Huth, Robert; Mielke, Alexander; Rehberg, Joachim; Winkler, MichaelLiteraturverz.
- ItemBulk-surface electro-thermodynamics and applications to electrochemistry(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 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 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.
- 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.
- ItemA compressible mixture model with phase transition(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) Dreyer, Wolfgang; Giesselmann, Jan; Kraus, ChristianeWe introduce a new thermodynamically consistent diffuse interface model of AllenCahn/NavierStokes type for multi-component flows with phase transitions and chemical reactions. For the introduced diffuse interface model, we investigate physically admissible sharp interface limits by matched asymptotic techniques. We consider two scaling regimes, i.e. a non-dissipative and a dissipative regime, where we recover in the sharp interface limit a generalized Allen-Cahn/Euler system for mixtures with chemical reactions in the bulk phases equipped with admissible interfacial conditions. The interfacial conditions satify, for instance, a YoungLaplace and a Stefan type law.
- ItemContinuum thermodynamics of chemically reacting fluid mixtures(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) Bothe, Dieter; Dreyer, WolfgangWe consider viscous and heat conducting mixtures of molecularly miscible chemical species forming a fluid in which the constituents can undergo chemical reactions. Assuming a common temperature for all components, a first main aim is the derivation of a closed system of partial mass and partial momentum balances plus a common balance of internal energy. This is achieved by careful exploitation of the entropy principle which, in particular, requires appropriate definitions of absolute temperature and chemical potentials based on an adequate definition of thermal energy that excludes diffusive contributions. The latter is crucial in order to obtain a closure framework for the interaction forces between the different species. The interaction forces split into a thermo-mechanical and a chemical part, where the former turns out to be symmetric if binary interactions are assumed. In the non-reactive case, this leads to a system of Navier-Stokes type sub-systems, coupled by interspecies friction forces. For chemically reacting systems and as a new result, the chemical interaction force is identified as a contribution which is non-symmetric, unless chemical equilibrium holds. The theory also provides a rigorous derivation of the so-called generalized thermodynamic driving forces, avoiding the use of approximate solutions to the Boltzmann equations which is common in the engineering literature. Moreover, starting with a continuum thermodynamic field theory right away, local versions of fundamental relations known from thermodynamics of homogeneous systems, like the Gibbs-Duhem equation, are derived. Furthermore, using an appropriately extended version of the entropy principle and introducing cross-effects already before closure as entropy invariant couplings between principal dissipative mechanisms, the Onsager symmetry relations are a strict consequence. With a classification of the factors forming the binary products in the entropy production according to their parity instead of the classical distinction between so-called fluxes and driving forces, the apparent anti-symmetry of certain couplings is thereby also revealed. If the diffusion velocities are small compared to the speed of sound, the well-known Maxwell-Stefan equations together with the so-called generalized thermodynamic driving forces follow in the special case without chemical reactions, thereby neglecting wave phenomena in the diffusive motion. This results in a reduced model having only the constituents mass balances individually. In the reactive case, this approximation via a scale separation argument is no longer possible. Instead, we first employ the partial mass and mixture internal energy balances, common to both model classes, to identify all constitutive quantities. Combined with the concept of entropy invariant model reduction, leaving the entropy production unchanged under the reduction from partial momentum balances to a single common mixture momentum balance, the chemical interactions yield an additional contribution to the transport coefficients, leading to an extension of the Maxwell-Stefan equations to chemically active mixtures. Within the considered model class for reactive fluid mixtures the new results are achieved for arbitrary free energy functions.
- ItemDetermination of stiffness and higher gradient coefficients by means of the embedded atom method: An approach for binary alloys(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2006) Boehme, Thomas; Dreyer, Wolfgang; Müller, Wolfgang H.For a quantitative theoretical description of phase separation and coarsening reliable data of stiffness constants and the so called Higher Gradient Coefficients (HGCs) are required. For that reason pair potentials of the Lennard-Jones type were used in [1] to provide a theoretical tool for their quantitative determination. Following up on this work these quantities are now calculated by means of the Embedded-Atom Method (EAM), a recently developed approach to describe interatomic potentials in metals. This is done, first, to achieve a better agreement between predicted and experimentally observed stiffness data as well as to avoid artifacts, such as the Cauchy paradox, and, second, to increase the trustworthiness of the HGCs for which experimental data are rarely available. After an introduction to the fundamentals of EAM it is outlined how it can be used for calculating stiffness constants and HGCs. In particular, Johnson's modification of EAM for nearest neighbor interactions [3] is applied to present explicit numerical results for a case study alloy, Ag-Cu, which has a simpleface-centered-cubic crystal structure and where it is comparatively easy to obtain all the required analysis data from the literature and to experimentally compare the predictions of mechanical data.
- 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.
- ItemThe equilibria of vapour-liquid systems revisited(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2007) Dreyer, Wolfgang; Kraus, ChristianeWe study equilibrium conditions of liquid-vapour phase transitions for a single substance at constant temperature. The phase transitions are modelled by a classical sharp interface model with boundary contact energy. We revisit this old problem mainly for the following reasons. Equilibria in a two-phase system can be established either under fixed external pressure or under fixed total volume. These two different settings lead to distinct equilibria, a fact that is usually ignored in the literature. In nature and in most technical processes, the approach of a two-phase system to equilibrium runs at constant pressure, whereas mathematicians prefer to study processes in constant domains, i.e. at constant volume. Furthermore, in the literature the sharp interface of the liquid and the vapour phase is usually described by a surface with high symmetry like a plane interface or a radially symmetric interface which has the shape of the boundary of a ball. In this paper we establish equilibrium conditions for pressure control as well as for volume control with arbitrary shapes of the interface. The results are derived by methods of differential geometry. Further, the common features and differences of pressure and volume control are worked out for some simple cases.
- ItemExact solutions to the Riemann problem for compressible isothermal Euler equations for two phase flows with and without phase transition(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2011) Dreyer, Wolfgang; Hantke, Maren; Warnecke, GeraldWe consider the isothermal Euler equations with phase transition between a liquid and a vapor phase. The mass transfer is modeled by a kinetic relation. We prove existence and uniqueness results. Further, we construct the exact solution for Riemann problems. We derive analogous results for the cases of initially one phase with resulting condensation by compression or evaporation by expansion. Further we present numerical results for these cases. We compare the results to similar problems without phase transition.
- 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.
- 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.
- 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.
- 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.
- 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.
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