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- 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 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.
- 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.
- 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.
- 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.
- ItemPhase transition and hysteresis in a rechargeable lithium battery revisited(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2009) Dreyer, Wolfgang; Gaberscek, Miran; Guhlke, Clemens; Huth, Robert; Jamnik, JankoWe revisit a model which describes the evolution of a phase transition that occurs in the cathode of a rechargeable lithium battery during the process of charging/discharging. The model is capable to simulate hysteretic behavior of the voltage
- ItemA model for the austenite-ferrite phase transition in steel including misfit stress(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2008) Dreyer, Wolfgang; Hömberg, Dietmar; Petzold, ThomasWe present a thermodynamically consistent model to describe the austenite-ferrite phase transition in steel. We consider the influence of the mechanical displacement field due to eigenstrains caused by volumetric expansion. The model equations are derived in a systematical framework. They are based on the conservation laws for mass and momentum and the second law of thermodynamics. By means of numerical computations for a simplified interface controlled model, we examine the influence of the mechanical contributions to the transformation kinetics and the equilibrium states.
- ItemWave trains, solitons and modulation theory in FPU chains(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2006) Dreyer, Wolfgang; Herrmann, Michael; Rademacher, Jens D.M.We present an overview of recent results concerning wave trains, solitons and their modulation in FPU chains. We take a thermodynamic perspective and use hyperbolic scaling of particle index and time in order to pass to a macroscopic continuum limit. While strong convergence yields the well-known p-system of mass and momentum conservation, we generally obtain a weak form of it in terms of Young measures. The modulation approach accounts for microscopic oscillations, which we interpret as temperature, causing convergence only in a weak, average sense. We present the arising Whitham modulation equations in a thermodynamic form, as well as analytic and numerical tools for the resolution of the modulated wave trains. As a prototype for the occurrence of temperature from oscillation-free initial data, we discuss various Riemann problems, and the arising dispersive shock fans, which replace Lax-shocks. We predict scaling and jump conditions assuming a generic soliton at the shock front.
- ItemMean field diffusion models for precipitation in crystalline GaAs including surface tension and bulk stresses(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2009) Dreyer, Wolfgang; Kimmerle, Sven-JoachimBased on a thermodynamically consistent model for precipitation in gallium arsenide crystals including surface tension and bulk stresses by Dreyer and Duderstadt, we propose different mathematical models to describe the size evolution of liquid droplets in a crystalline solid. The first class of models treats the diffusion-controlled regime of interface motion, while the second class is concerned with the interface-controlled regime of interface motion. Our models take care of conservation of mass and substance. We consider homogenised models, where different length scales of the experimental situation have been exploited in order to simplify the equations. These homogenised models generalise the well-known Lifshitz-Slyozov-Wagner model for Ostwald ripening. Mean field models capture the main properties of our system and are well adapted for numerics and further analysis. Numerical evidence suggests in which case which one of the two regimes might be appropriate to the experimental situation.
- ItemOn unwanted nucleation phenomena at the wall of a VGF chamber(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2008) Dreyer, Wolfgang; Duderstadt, Frank; Eichler, Stefan; Naldzhieva, MargaritaThis is preliminary study on a phenomenon that happens during crystal growth of GaAs in a vertical gradient freeze (VGF) device. Here unwanted polycrystals nucleate at the chamber wall and move into the interior of the crystal. This happens within an undercooled region in the vicinity of the triple point, where the liquid-solid interface meets the chamber wall. The size and shape of that region is modelled by the Gibbs-Thomson law, which will be rederived in this paper. Hereafter we identify the crucial parameter, whose proper adjustment may minimize the undercooled region.