Dissertations / Theses on the topic 'Phase-field model'

We address two problems in this thesis. First, a phase-field model for structural phase transformations in solids and second, a model for dynamic fracture. The existing approaches for both phase transformations and fracture can be grouped into two categories. Sharp-interface models, where interfaces are singular surfaces; and regularized-interface models, such as phase-field models, where interfaces are smeared out. The former are challenging for numerical solutions because the interfaces or crack needs to be explicitly tracked, but have the advantage that the kinetics of existing interfaces or cracks and the nucleation of new interfaces can be transparently and precisely prescribed. The diffused interface models such as phasefield models do not require explicit tracking of interfaces and makes them computationally attractive. However, the specification of kinetics and nucleation is both restrictive and extremely opaque in such models. This prevents straightforward calibration of phase-field models to experiment and/or molecular simulations, and breaks the multiscale hierarchy of passing information from atomic to continuum. Consequently, phase-field models cannot be confidently used in dynamic settings. We present a model which has all the advantages of existing phase-field models but also allows us to prescribe kinetics and nucleation criteria. We present a number of examples to characterize and demonstrate the features of the model. We also extend it to the case of multiple phases where preserving kinetics of each kind of interface is more complex. We use the phase transformation model with certain changes to model dynamic fracture. We achieve the advantage of prescribing nucleation and kinetics independent of each other. We demonstrate examples of anisotropic crack propagation and crack propagation on an interface in a composite material. We also report some limitations of phase-field models for fracture which have not been mentioned in the existing literature. These limitations include dependence of effective crack width and hence the effective surface energy on the crack speed, lack of a reasonable approximation for the mechanical response of cracked region and inability to model large deformations.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Chemical Engineering, 2008.
Includes bibliographical references (p. 261-270).
Oxygen precipitate caused by oxygen supersaturation is the most common and important defects in Czochralski (CZ) silicon. The presence of oxygen precipitate in silicon wafer has both harmful and beneficial effects on the microelectronic device production. Oxygen precipitates are useful for gathering metallic contaminants away from the device regions and for increasing the mechanical strength of the wafer [Borghesi, 1995], but they also can destroy the electrical and mechanical characteristics of the semiconductor and microelectronic devices [Abe, 1985; Kolbesen, 1985]. The understanding of the mechanism of the formation and growth of the oxygen precipitates in CZ silicon is a key to improve the quality of silicon wafer. The goal of this thesis is to provide a full understanding of the growth of an isolated oxygen precipitate in CZ silicon and its morphological evolution by means of phase-field method, and to gain the insight of the morphological transition of the oxygen precipitate and the distribution of oxygen, vacancy, and self-interstitial around the single oxygen precipitate. The traditional approach to simulate multiphase system is the sharp interface model. Sharp interface model requires tracking the interface between phases, which make the simulation much difficult and complicate. Phase-field model offers an alternative approach for predicting mesoscale morphological and microstructure evolution of inhomogeneous multiphase system. The most significant computational advantage of a phase-field model is that explicit tracking of the interface is unnecessary. In this thesis, the phase-field model is applied to simulate the evolution of oxygen precipitates in CZ silicon. A phase-field model for a two-component inhomogeneous system was first derived to set up the framework of phase-field method and a dynamically adaptive finite element method also was built to specifically solve phase-field equations. This model was used to investigate the effects of interfacial and elastic properties on the growth of a single precipitate, coarsening of two precipitates, and competitive growth of multiple precipitates. For an isolated precipitate growth, both elastic energy and interfacial energy affect the precipitate morphological evolution.
(cont.) Numerical results show the shape of the precipitate is determined by the relative contributions of elastic energy and interfacial energy, the degree of elastic anisotropy, and the degree of interfacial anisotropy. A dimensionless length scale LS3 was defined to represent the relative contributions of the interfacial energy and elastic energy. For large LS3 (LS3 > 5), the anisotropic elasticity plays a dominant role and precipitate evolves to held the elastic anisotropy even if the interfacial anisotropy is very strong. However, if LS3 ~1 or elasticity is isotropic, the strong anisotropy ([epsilon]4 =/> 0.05 ) of the interface will be the dominant factor to determine the precipitate shape. The growth rate of an isolated precipitate follows the diffusion-controlled power law. The elasticity significantly decreases the precipitate growth rate, while the anisotropy of the interface does not. Coarsening of two precipitates was also explored with different interfacial and elastic properties. The results also show that both elasticity and interfacial anisotropy enhance the coarsening rate. For competitive growth of multiple precipitates, a gap was found to be developed between the precipitates because of the precipitate screening, but this gap could be destroyed by increasing the interfacial energy or introducing elastic energy. Based on the framework of the previous phase-field model, another phase-field model coupling CALPHAD thermodynamic assessment was developed to simulate the growth of the oxygen precipitate in CZ isilicon. An asymptotic analysis was performed to understand the phase-field model at the sharp interface limit and all physical principles of the solid precipitate growth problem were recovered. a Cristobalite and amorphous oxygen precipitates were calculated at different orientations and temperatures. Disk-like shape, square, ellipse, a slightly deformed sphere are reproduced for oxygen precipitates, which agrees with the experimental observations very well. In addition, the growth rates of amorphous precipitates and a cristobalite precipitates at different temperatures show that at high temperature 1100 °C, amorphous precipitate has the largest growth rate, while at low temperature 900 °C, a cristobalite precipitate grows faster.
(cont.) This qualitatively explained why different polymorphs and shapes of the oxygen precipitate were observed in experiments at different annealing temperatures.
by Minggang She.
Ph.D.

Thesis (Ph. D.)--University of Kentucky, 2006.
Title from document title page (viewed on January 25, 2007). Document formatted into pages; contains: xiii, 162 p. : ill. (some col.). Includes abstract and vita. Includes bibliographical references (p. 151-157).

Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mechanical Engineering, 2018.
Cataloged from PDF version of thesis.
Includes bibliographical references (pages 117-122).
Interfaces, such as grain boundaries, solid-liquid interfaces and solid-solid heterophase interfaces, are important features found in materials. Material properties such as fracture toughness, corrosion susceptibility and high temperature creep are influenced by grain boundary physics. The structure of grain boundaries affects their properties. In this thesis, we have developed a predictive model for a particular grain boundary structure-property relationship: the permeation of liquid gallium through aluminum grain boundaries. Liquid gallium is known to permeate through aluminum grain boundaries. The reduction in interface energy by the replacement of one Al-Al grain boundary interface with two Al-Ga interfaces drives the permeation. The speed of permeation depends on factors which affect the Al-Al grain boundary energy, such as grain boundary crystallography, applied stress, and temperature. Literature suggests two major hypotheses for the permeation mechanism: front propagation, and diffusion and coalescence. We have used phase field methods to develop a predictive model for the permeation of gallium through individual aluminum grain boundaries. The model uses location dependent grain boundary energy (LDGBE) distributions for aluminum grain boundaries to predict permeation velocities. Importantly, by changing the model's parameters, its behavior can be adjusted smoothly from front propagation, to diffusion and coalescence. We have used experimental data collected by Hugo and Hoagland, along with LDGBE maps computed by our collaborators, to infer the parameters of the phase field model. The inference has been done in a Bayesian framework, which gives us estimates of the model parameters with quantifiable uncertainty. The inferred model parameters strongly support the front propagation hypothesis. We discuss the implications of this inference and potential limitations of our methodology.
by Raghav Aggarwal.
Ph. D.

The objective of this paper is to propose a 2-D time-independent phase-field model with validating its performance as well as applying it for simulating existing representative experiments. Firstly, the section of the literature review provides an overview of quasi-brittle material and brittle fracture behaviours, as well as the existing FE models from both discontinuous and continuous approaches for simulating fracture behaviours. Next, the governing equations of the proposed phase-field model are determined, which are based on traditional Griffith’s theory as well as a specific variational method evolved from that. The proposed model is implemented in Abaqus. In particular, the implementation is achieved by using the User Subroutine in order to take the phase-field into account. The proposed model is validated by simulating a pure-tension and a pure shear test. In this part, not only the effect of discretisation but also the effects of length parameter and energy release rate has been discussed, of which the latter effect is exclusive in phase-field method. Finally, the validated model is used for simulating two sets of existing experiments, including a mixed-mode test and a series of Brazilian disks test. The results in both validation and simulation part indicate that the proposed model can successfully simulate both crack initiation and propagation in these cases, and good qualitative agreement with theoretical or experimental results can be observed.

Thesis (Ph.D.)--George Mason University, 2008.
Vita: p. 203. Thesis director: Thomas Wanner. Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Computational Sciences and Informatics. Title from PDF t.p. (viewed Aug. 28, 2008). Includes bibliographical references (p. 199-202). Also issued in print.

The research conducted at the department of Electrical Engineering of the University of South Florida campus in Tampa only covers the electrical aspect of electric drives. However, the performance of electric machinery is significantly impacted by temperature variation. The literature review shows three main control techniques in use today in electric drives namely, Scalar control, Direct Torque control and Field Oriented control. This thesis presents a temperature rise of rotor bars, stator winding, stator core and stator frame in a running three phase field-oriented controlled induction machine. A literature search shows that none of research has been carried out to investigate a thermal response of a field-oriented controlled induction motor. With this motivation, we were able to implement a lumped parameters thermal model of a three-phase field-oriented IM in MATLAB Simulink, which allows us to determine that rotor bars have the highest temperatures rising to 84 degrees Celsius. This confirms that rotors bars are the hottest part of a running IM as stipulated in literature.

In dieser Arbeit werden Zufallsfeld-Ising-Modelle mit einem eingefrorenen dichotomen symmetrischen Zufallsfeld für den eindimensionalen Fall und das Bethe-Gitter untersucht. Dabei wird die kanonische Zustandssumme zu der eines einzelnen Spins in einem effektiven Feld umformuliert. Im ersten Teil der Arbeit werden das mulktifraktale Spektrum dieses effektiven Feldes untersucht, Übergänge im Spektrum erklärt und Ungleichungen zwischen lokalen und globalen Dimensionsbegriffen bewiesen, die eine weitgehend vollständige Charakterisierung des multifraktalen Spektrums durch eine Reihe von Schranken erlauben. Ein weiterer Teil der Arbeit beschäftigt sich mit einer ähnlichen Charakterisierung des Maßes der lokalen Magnetisierung, das aus dem Maß des effektiven Feldes durch Faltung hervorgeht. In diesem Zusammenhang wird die Faltung von Multifraktalen in einem allgemeineren Rahmen behandelt und Zusammenhänge zwischen den multifraktalen Eigenschaften der Faltung und denen der gefalteten Maße bewiesen. Im dritten Teil der Dissertation wird der Phasenübergang von Ferro- zu Paramagnetismus im Modell auf dem Bethe Gitter untersucht. Neben verbesserten exakten Schranken für die Eindeutigkeit des paramagnetischen Zustands werden im wesentlichen drei Kriterien für die tatsächliche Lage des Übergangs angegeben und numerisch ausgewertet. Die multifraktalen Eigenschaften des effektiven Felds im Modell auf dem Bethe-Gitter schließlich erweisen sich als trivial, da die interessanten Dimensionen nicht existieren
In this work random field Ising models with quenched dichotomous symmetric random field are considered for the one-dimensional case and on the Bethe lattice. To this end the canonical partition function is reformulated to the partition function of one spin in an effective field. In the first part of the work the multifractal spectrum of this effective field is investigated, transitions in the spectrum are explained and inequalities between local and global generalized fractal dimensions are proven which allow to characterize the multifractal spectrum bei various bounds. A further part of the work is dedicated to the characterization of the measure of the local magnetization which is obtained by convolution of the measure of the effective field with itself. In this context the convolution of multifractals is investigated in a more general setup and relations between the multifractal properties of the convolution and the multifractal properties of the convoluted measures are proven. The phase transition from ferro- to paramagnetismus for the model on the Bethe lattice is investigated in the third part of the thesis. Apart from improved exact bounds for the uniqueness of the paramagnetic state essentially three criteria for the transition are developped and numerically evaluated to determine the transition line. The multifractal properties of the effective field for the model on the Bethe lattice finally turn out to be trivial because the interesting dimensions do not exist

The diffuse-interface phase-field model is a powerful method to simulate and predict mesoscale microstructure evolution in materials using fundamental properties of thermodynamics and kinetics. The objective of this dissertation is to develop phase-field model for simulation and prediction of interdiffusion behavior and evolution of microstructure in multiphase binary and ternary systems under composition and/or temperature gradients. Simulations were carried out with emphasis on multicomponent diffusional interactions in single-phase system, and microstructure evolution in multiphase systems using thermodynamics and kinetics of real systems such as Ni-Al and Ni-Cr-Al. In addition, selected experimental studies were carried out to examine interdiffusion and microstructure evolution in Ni-Cr-Al and Fe-Ni-Al alloys at 1000°C. Based on Onsager’s formalism, a phase-field model was developed for the first time to simulate the diffusion process under an applied temperature gradient (i.e., thermotransport) in single- and two-phase binary alloys. Development of concentration profiles with uphill diffusion and the occurrence of zeroflux planes were studied in single-phase diffusion couples using a regular solution model for a hypothetical ternary system. Zero-flux plane for a component was observed to develop for diffusion couples at the composition that corresponds to the activity of that component in one of the terminal alloys. Morphological evolution of interphase boundary in solid-to-solid two-phase diffusion couples (fcc-γ vs. B2-β) was examined in Ni-Cr-Al system with actual thermodynamic data and concentration dependent chemical mobility. With the instability introduced as a small initial compositional fluctuation at the interphase boundary, the evolution of the interface morphology was found to vary largely as a function of terminal alloys and related composition dependent chemical mobility. In a binary Ni-Al system, multiphase diffusion couples of fcc-γ vs. L12-γ′, γ vs. γ+γ′ and γ+γ′ vs. γ+γ′ were simulated with alloys of varying compositions and volume fractions of second phase (i.e., γ′). Chemical mobility as a function of composition was employed in the study with constant gradient energy coefficient, and their effects on the final interdiffusion microstructure was examined. Interdiffusion microstructure was characterized by the type of boundaries formed, i.e. Type 0, Type I, and Type II boundaries, following various experimental observations in literature and thermodynamic considerations. Volume fraction profiles of alloy phases present in the diffusion couples were measured to quantitatively analyze the formation or dissolution of phases across the boundaries. Kinetics of dissolution of γ′ phase was found to be a function of interdiffusion coefficients that can vary with composition and temperature. The evolution of interdiffusion microstructures in ternary Ni-Cr-Al solid-to-solid diffusion couples containing fcc-γ and γ+β (fcc+B2) alloys was studied using a 2D phase-field model. Alloys of varying compositions and volume fractions of the second phase (β) were used to simulate the dissolution kinetics of the β phase. Semi-implicit Fourier-spectral method was used to solve the governing equations with chemical mobility as a function of compositions. The simulation results showed that the rate of dissolution of the β phase (i.e., recession of β+γ twophase region) was dependent on the composition of the single-phase γ alloy and the volume fraction of the β phase in the two-phase alloy of the couple. Higher Cr and Al content in the γ alloy and higher volume fraction of β in the γ+β alloy lower the rate of dissolution. Simulated results were found to be in good agreement with the experimental observations in ternary Ni-Cr- Al solid-to-solid diffusion couples containing γ and γ+β alloys. For the first time, a phase-field model was developed to simulate the diffusion process under an applied temperature gradient (i.e., thermotransport) in multiphase binary alloys. Starting from the phenomenological description of Onsager’s formalism, the field kinetic equations are derived and applied to single-phase and two-phase binary system. Simulation results show that a concentration gradient develops due to preferential movement of atoms towards the cold and hot end of an initially homogeneous single-phase binary alloy subjected to a temperature gradient. The temperature gradient causes the redistribution of both constituents and phases in the two-phase binary alloy. The direction of movement of elements depends on their atomic mobility and heat of transport values.
Ph.D.
Department of Mechanical, Materials and Aerospace Engineering;
Engineering and Computer Science
Materials Science & Engr PhD

This work examines the fea.'libility of a method for the multi-phase segmentation of colour images and gives numerical examples of the results. The method combines the vector-valued Allen-Cahn (phase field) equation with initial data fitting terms, which are closely related to the MumfordShah problem and the level set segmentation by Chan and Vese. The numerical solution is performed using a multigrid algorithm over a set of nested finite element spaces, thereby producing an efficient and robust method for the segmentation of large images. A priori error estimates are included. Tests are performed to confirm the efficiency of the multigrid scheme, and salient examples of the results are provided.

The ubiquity of free boundary problems in biology, with which we are mainly concerned, led us to adopt a mathematical technique to render them tractable and relatively easy to solve. The phase field method, used when dealing with dynamical systems with moving boundary condition, addresses arising difficulties involved with tracking those boundaries. In this work we model the motion of neutrophils, cells of the immune system, in response to chemical driving. Their morphology changes dynamically as they move to neutralize their target, a mechanism called chemotaxis. Additionally, we propose a model for neural cell growth, which is insti- gated by mediators, a process termed chemotropism. When coupled to an internal mechanism of actin polymerization, this induces the advancement of the axonal tip. Lastly, we used the method to build a generalized model from which both chemo- tactic and chemotropic models can be derived. These three problems were solved by the construction of free energy functionals F that captured the main features of the dynamics, in relation to the order parameters φ that distinguished the phases of the system as well as their interfaces. We considered the membrane of the cells, their inside and outside, as well as their leading edges. The governing partial dif- ferential equations were obtained by a variational differentiation of F with respect to the fields. Following this method, we were able to model cell morphodynamics in two and three dimensions. The major contribution of our work lies in the reduction of the complexity of those problems: we solve partial differential equations of fields coupled to the underlying dynamics at the molecular level, which are derived from a closed form generalized functional describing both the cell motion, deformation and growth.
L'abondance des problèmes à interface libre dans la biologie auxquels nous sommes principallement interéssés, nous a conduits à adopter une technique qui les rend relativement faciles à resoudre. Phase Field Method, utilisée pour examiner des systèmes dynamiques possédants des conditions aux limites en mouvement, resoud la diffuculté provenant de suivre leur évolution. Dans ce travail, on modèle le mouvement des neutrophiles qui sonts des cellules du système immunitaire en réponse aux signaux chimiques. Leur morphologie change dynamiquement quand ils se deplancent pour neutraliser leur cible: ce mécanisme est appelé chimiotaxie. Egalement, on propose un modèle pour le développement des cellules nerveuses induit par des médiateurs. Ce processus est nommé chémotropisme. Ce dernier, quand il est lié au mécanisme interne responsable de la polymérisation de l'actine, induit l'avancement du bout de l'axone. Finalement, on a employé la méthode pour construire un modèle généralisé qui permet de dériver les deux modèles chimiotactique et chémotropique. Ces problèmes ont été resouds en construisant des fonctionelles d'énergie libre F , capturant les caractéristiques principaux de la dynamique en fonction d'un paramètre d'order φ qui permet de dinstinguer les différentes phases du système aussi bien que les interfaces qui les séparent. Les éequations aux dérivées partielles décrivant leur evolution sont déterminées en effectuant une différenciation variationelle de F par rapport au champs φ. En suivant cette méthode, on était capable de reproduire la dynamique des morphologies en deux et trois dimensions. La contribution majeure de notre travail réside dans la réduction de la complexité de ces problèmes en suivant les équations aux dérivées partielles. Ces dernières sont liées aux mécanismes internes au niveau moléculaire dérivés dune fonctionelle généralisée F qui décrit le mouvement de la cellule ainsi que sa déformation et sa croissance.

This thesis examines the Heisenberg antiferromagnetic spin chain in one dimension (1D) with a crystal field splitting term and applied magnetic field term. We use theoretical techniques from quantum field theory and conformal field theory (CFT) to make predictions about the excitation spectrum for our model. We then use Density Matrix Renormalization Group (DMRG) numerical techniques to simulate our spin chain and extract the energy spectrum as we vary our crystal field splitting and magnetic field terms. These results are compared and we examine where theoretical calculations accurately describe our system. This work is motivated by recent experimental work done on SrNi₂Vi₂O₈ by Bera et al. [1] which is a quasi-1D material with weakly coupled spin chains in the bulk. These 1D chains are expected to be described by the Hamiltonian we study in this thesis, and we neglect interchain coupling. We first consider our system where the crystal field splitting term is set to zero, which can be described theoretically using a mapping to the non linear sigma model (NLSM). Near the critical field, it undergoes a Bose condensation transition whose excitation spectrum can be mapped to non-interacting fermions in 1D. We then consider large negative crystal field splitting, and find that near small applied magnetic field we can describe some excited states using Landau-Ginsburg theory. Near critical field, we show that the transition is in the Ising universality, and use results from CFT to predict the spectrum for finite size systems. This allows us to make predictions about where the transition field would be for very large or infinite system size. Finally, we examine our crystal field splitting tuned to the value obtained in Ref. 1, which is a small, negative value. We observe qualitative elements in this spectrum from the spectra obtained at zero and large negative crystal field splitting.
Science, Faculty of
Physics and Astronomy, Department of
Graduate

Thesis: Ph. D., Massachusetts Institute of Technology, Department of Nuclear Science and Engineering, 2018.
This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.
Cataloged from student-submitted PDF version of thesis.
Includes bibliographical references (pages 389-400).
Zircaloy-4 has been used in pressurized water reactors (PWRs) for decades, and enhanced corrosion rates in reactors compared to out-of-pile have long been observed. However, the exact mechanism explaining the early departure from autoclave kinetics after 3-5 microns of oxide have formed has proved elusive. This thesis considers and evaluates a number of possible explanations for this early acceleration in kinetics. The bulk of the evidence points to Fe depletion from secondary phase particles (SPPs) as the culprit in enhancing Zircaloy-4 corrosion rates in PWRs. These new findings have been incorporated in a mechanistic finite-element phase field model of Zircaloy-4 corrosion called HOGNOSE. It accounts for both diffusion-and drift-based oxygen anion transport in Zircaloy-4 by including the effects of radiation-induced evolution of SPPs in changing the contribution of a local charge transport inequality through their depletion and release of iron. By addressing the imbalance in charged particle transport, the code can be adapted to model multiple zirconium-based alloys in autoclave and irradiated conditions with minimal parameter fitting. Rather than the typical empirical approach, HOGNOSE uses a physics-based methodology to capture the early agreement between autoclave and in-reactor data and the point at which reactor kinetics are enhanced compared to autoclave kinetics. HOGNOSE results agree fairly well with those observed in experiments for oxide thicknesses less than 10 microns, above which other enhancement mechanisms can no longer be safely ignored. HOGNOSE captures increasing amorphization with decreasing temperature, and more subtle corrosion rate enhancement at higher temperatures. Comparisons between HOGNOSE results and literature data suggest that the next focus for mechanistic modeling should consider additional neutron flux effects. To support HOGNOSE development, corrosion testing of Zircaloy-4 in steam at atmospheric pressure and 415 degrees Celsius was performed. Samples were analyzed using a focused ion beam/scanning electron microscope (FIB/SEM) to obtain oxide thickness measurements with greater temporal resolution than is widely provided by autoclave testing. Oxide thickness data was used to determine the thermal dependence of oxygen diffusivity in the oxide within HOGNOSE. HOGNOSE would also benefit from measurements of the concentrations and charge states of cation dopants in post-irradiated Zircaloy oxides to help determine whether this model is truly accurate in its physical description.
by Andrew Frederic Dykhuis.
Ph. D.

Les simulations numériques des fissures fragiles par les modèles d’endommagement à gradient deviennent main- tenant très répandues. Les résultats théoriques et numériques montrent que dans le cadre de l’existence d’une pre-fissure la propagation suit le critère de Griffith. Alors que pour le problème à une dimension la nucléation de la fissure se fait à la contrainte critique, cette dernière propriété dimensionne le paramètre de longueur interne.Dans ce travail, on s’attarde sur le phénomène de nucléation de fissures pour les géométries communément rencontrées et qui ne présentent pas de solutions analytiques. On montre que pour une entaille en U- et V- l’initiation de la fissure varie continument entre la solution prédite par la contrainte critique et celle par la ténacité du matériau. Une série de vérifications et de validations sur diffèrent matériaux est réalisée pour les deux géométries considérées. On s’intéresse ensuite à un défaut elliptique dans un domaine infini ou très élancé pour illustrer la capacité du modèle à prendre en compte les effets d’échelles des matériaux et des structures.Dans un deuxième temps, ce modèle est étendu à la fracturation hydraulique. Une première phase de vérification du modèle est effectuée en stimulant une pré-fissure seule par l’injection d’une quantité donnée de fluide. Ensuite on étudie la simulation d’un réseau parallèle de fissures. Les résultats obtenus montrent qu’il a qu’une seule fissure qui se propage et que ce type de configuration minimise mieux l’énergie la propagation d’un réseau de fractures. Le dernier exemple se concentre sur la stabilité des fissures dans le cadre d’une expérience d’éclatement à pression imposée pour l’industrie pétrolière. Cette expérience d’éclatement de la roche est réalisée en laboratoire afin de simuler les conditions de confinement retrouvées lors des forages.La dernière partie de ce travail se concentre sur la rupture ductile en couplant le modèle à champ de phase avec les modèles de plasticité parfaite. Grâce à l’approche variationnelle du problème on décrit l’implantation numérique retenue pour le calcul parallèle. Les simulations réalisées montrent que pour une géométrie légèrement entaillée la phénoménologie des fissures ductiles comme par exemple la nucléation et la propagation sont en concordances avec ceux reportées dans la littérature
Phase-field models, sometimes referred to as gradient damage, are widely used methods for the numerical simulation of crack propagation in brittle materials. Theoretical results and numerical evidences show that they can predict the propagation of a pre-existing crack according to Griffith’s criterion. For a one- dimensional problem, it has been shown that they can predict nucleation upon a critical stress, provided that the regularization parameter is identified with the material’s internal characteristic length.In this work, we draw on numerical simulations to study crack nucleation in commonly encountered geometries for which closed-form solutions are not available. We use U- and V-notches to show that the nucleation load varies smoothly from the one predicted by a strength criterion to the one of a toughness criterion when the strength of the stress concentration or singularity varies. We present validation and verification of numerical simulations for both types of geometries. We consider the problem of an elliptic cavity in an infinite or elongated domain to show that variational phase field models properly account for structural and material size effects.In a second movement, this model is extended to hydraulic fracturing. We present a validation of the model by simulating a single fracture in a large domain subject to a control amount of fluid. Then we study an infinite network of pressurized parallel cracks. Results show that the stimulation of a single fracture is the best energy minimizer compared to multi-fracking case. The last example focuses on fracturing stability regimes using linear elastic fracture mechanics for pressure driven fractures in an experimental geometry used in petroleum industry which replicates a situation encountered downhole with a borehole called burst experiment.The last part of this work focuses on ductile fracture by coupling phase-field models with perfect plasticity. Based on the variational structure of the problem we give a numerical implementation of the coupled model for parallel computing. Simulation results of a mild notch specimens are in agreement with the phenomenology of ductile fracture such that nucleation and propagation commonly reported in the literature

We consider fully practical finite element approximations of the nonlinear parabolic Cahn-Hilliard system [Mathematical equation appears here. To view, please open pdf attachment] subject to an initial condition u0(.) ∈ [−1, 1] on the conserved order parameter u ∈ [−1, 1], and mixed boundary conditions. Here γ ∈ R>0 is the interfacial parameter, [Symbol appears here. To view, please open pdf attachment]∈ R>0 is a time-scaling parameter, α ∈ R≥0 is the field strength parameter, [Symbol appears here. To view, please open pdf attachment] is the obstacle potential, and c(x, u) and b(x, u) are the diffusion coefficients. Furthermore, w is the chemical potential, φ is the electro-static potential and {v, p} are the velocity and pressure. The system, in the limit γ → 0, models the evolution of an unstable interface between two dielectric media in the presence of an electric field, which is quasi-static, and a Stokes flow for the dielectric media. Our goal is to produce stable fully practical finite element approximations to the phase field model above. Additionally, we would like to reproduce the morphologies observed in studies by Buxton and Clarke in [28], and Kim and Lu in [53, 56]. The presence of the electric field and kinetics should drive the interface growth. Initially restricting ourselves to the case without kinetics, we consider coupled and decoupled finite element approximations of the Cahn-Hilliard system. A coupled system is a non-linear algebraic system where the constituent systems are solved simultaneously at each time-step. A decoupled system splits the constituent systems so that they are solved separately and sequentially. Existence, stability and convergence results are presented for a coupled scheme and numerical results are given in two space dimensions. To develop a computationally efficient approximation we present a decoupled scheme with conditional stability in two space dimensions. Numerical results demonstrate that it is a suitable approximation to the coupled scheme. Introducing kinetics to the system requires the careful consideration of both the boundary conditions and mass conservation of the system. A modified coupled scheme admits existence, stability and convergence results. We investigate the applicability of several fast solution methods for the Stokes system. We also present evidence that the MINI-element for the velocity space is more computationally efficient than the Taylor-Hood element. Using further optimisation techniques, such as solving the Stokes system on a coarser mesh, we are able compute results in three dimensions efficiently. Numerous numerical results are presented in two and three dimensions.

We present a mean field model for a mixture of AB diblock-copolymers and A-block selective nanoparticles confined between two identical non-selective walls. A horizontally symmetric lamellar structure of the nanocomposite is considered where nanoparticles are allowed to segregate between the polymer–wall interfaces. For a fixed value of wall separation, we study changes in the free energy as a function of the number of lamellar layers and the amount of nanoparticle uptake in the A-phase denoted by y = ϕx with 0 ≤ x ≤ 1 for a given value of ϕ, where ϕ is the overall nanoparticle volume fraction. The absorption isotherm for nanoparticle uptake in the A-phase as a function of ϕ shows saturation beyond a threshold value ϕs, and the optimal value of uptake y increases with increasing strength of monomer–nanoparticle attractive interaction. Increasing ϕ above ϕs produces a decrease in the optimal number of lamellar layers which is related to a jump-like transition of the chain extension. The effect of varying film thickness is also studied. By considering A-block selective walls we also investigated a wetting transition of the copolymer film and found the transition to be discontinuous. A corresponding phase diagram is constructed
Dieser Beitrag ist mit Zustimmung des Rechteinhabers aufgrund einer (DFG-geförderten) Allianz- bzw. Nationallizenz frei zugänglich

We present a mean field model for a mixture of AB diblock-copolymers and A-block selective nanoparticles confined between two identical non-selective walls. A horizontally symmetric lamellar structure of the nanocomposite is considered where nanoparticles are allowed to segregate between the polymer–wall interfaces. For a fixed value of wall separation, we study changes in the free energy as a function of the number of lamellar layers and the amount of nanoparticle uptake in the A-phase denoted by y = ϕx with 0 ≤ x ≤ 1 for a given value of ϕ, where ϕ is the overall nanoparticle volume fraction. The absorption isotherm for nanoparticle uptake in the A-phase as a function of ϕ shows saturation beyond a threshold value ϕs, and the optimal value of uptake y increases with increasing strength of monomer–nanoparticle attractive interaction. Increasing ϕ above ϕs produces a decrease in the optimal number of lamellar layers which is related to a jump-like transition of the chain extension. The effect of varying film thickness is also studied. By considering A-block selective walls we also investigated a wetting transition of the copolymer film and found the transition to be discontinuous. A corresponding phase diagram is constructed.
Dieser Beitrag ist mit Zustimmung des Rechteinhabers aufgrund einer (DFG-geförderten) Allianz- bzw. Nationallizenz frei zugänglich.

Towing-tank experiments are performed for a surface combatant as it undergoes static and dynamic planar motion mechanism maneuvers in calm water. The data includes global forces/moment/motions and phase-averaged local flow-fields, and uncertainty assessment. The geometry is DTMB model 5512, which is a 1/46.6 scale geosym of DTMB model 5415, with L = 3.048 m. The experiments are performed in a 3.048 × 3.048 × 100 m towing tank. The measurement system features a custom designed planar motion mechanism, a towed stereoscopic particle image velocimetry system, a Krypton contactless motion tracker, and a 6-component loadcell. The forces/moment and UA are conducted in collaboration with two international facilities (FORCE and INSEAN), including test matrix and overlapping tests using the same model geometry but with different scales. Quality of the data is assessed by monitoring the statistical convergence, including tests for randomness, stationarity, and normality. Uncertainty is assessed following the ASME Standards (1998 and 2005). Hydrodynamic derivatives are determined from the forces/moment data by using the Abkowitz (1966) mathematical model, with two different 'Multiple-Run (MR)' and 'Single-Run (SR)' methods. The results for reconstructions of the forces/moment indicate that usually the MR method is more accurate than the SR. Comparisons are made of the hydrodynamic derivatives across different facilities. The scale effect is small for sway derivatives, whereas considerable for yaw derivatives. Heave, pitch, and roll motions exhibit cross-coupling between the motions and forces and moment data, as expect based on ship motions theory. Hydrodynamic derivatives are compared between different mount conditions. Linear derivatives values are less sensitive to the mounting conditions, whereas the non-linear derivatives are considerably different. Phase-averaged flowfield results indicate maneuvering-induced vortices and their interactions with the turbulent boundary layer. The tests are sufficiently documented and detailed so as to be useful as benchmark EFD data for CFD validation.

We present extensive testing in order to find the optimum balance among errors associated with time integration, spatial discretization, and splitting for a fully spectral semi implicit scheme of the phase field crystal model. The scheme solves numerically the equations of dissipative dynamics of the binary phase field crystal model proposed by Elder et al. [Elder et al, 2007]. The fully spectral semi implicit scheme uses the operator splitting method in order to decompose the complex equations in the phase field crystal model into sub-problems that can be solved more efficiently. Using the combination of non-trivial splitting with the spectral approach, the scheme leads to a set of algebraic equations of diagonal matrix form and thus easier to solve. Using this method developed by the BCAST research team we are able to show that it speeds up the computations by orders of magnitude relative to the conventional explicit finite difference scheme, while the costs of the pointwise implicit solution per timestep remains low. Comparing both the finite difference scheme used by Elder et al [Elder et al, 2007] to the spectral semi implicit scheme, we are also able to show that the finite differencing cannot compete with the spectral differencing in regards to accuracy. This is mainly due to numerical dissipation in finite differencing. In addition the results show that this method can efficiently be parallelized for distributed memory systems, where an excellent scalability with the number of CPUs. We have applied the semi-implicit spectral scheme for binary alloys to explore polycrystalline dendritic solidification. The kinetics of transformation has been analysed in terms of Johnson-Mehl-Avrami-Kolmogorov formalism. We show that Avrami plots are not linear, and the respective Avrami-Kolmogorov exponents (PAK) vary with the transformed fraction (or time). Using the semi-implicit spectral scheme we have been able to provide extensive numerical testing of methods in solving the single component case. This has been demonstrated by using unconditional time stepping with comparable simulations using conditional time stepping. We show the accuracy of the solution for unconditional time stepping is not compromised and furthermore computational efficiency can be significantly increased with the introduction of this scheme. Finally we have investigated how the composition of the initial liquid phase influences the eutectic morphology evolving during solidification. This is the first study that addresses this question using the dynamical density functional theory.

The investigation of failure in brittle materials, subjected to dynamic transient loading conditions, represents one of the ongoing challenges in the mechanics community. Progresses on this front are required to support the design of engineering components which are employed in applications involving extreme operational regimes. To this purpose, this thesis is devoted to the development of a framework which provides the capabilities to model how crack patterns form and evolve in brittle materials and how they affect the quantitative description of failure. The proposed model is developed within the context of diffusive interfaces which are at the basis of a new class of theories named phase field models. In this work, a set of additional features is proposed to expand their domain of applicability to the modelling of (i) rate and (ii) pressure dependent effects. The path towards the achievement of the first goal has been traced on the desire to account for micro-inertia effects associated with high rates of loading. Pressure dependency has been addressed by postulating a mode-of-failure transition law whose scaling depends upon the local material triaxiality. The governing equations have been derived within a thermodynamically-consistent framework supplemented by the employment of a micro-forces balance approach. The numerical implementation has been carried out within an updated lagrangian finite element scheme with explicit time integration. A series of benchmarks will be provided to appraise the model capabilities in predicting rate-pressure-dependent crack initiation and propagation. Results will be compared against experimental evidences which closely resemble the boundary value problems examined in this work. Concurrently, the design and optimization of a complimentary, improved, experimental characterization platform, based on the split Hopkinson pressure bar, will be presented as a mean for further validation and calibration.

Physics
Ph.D.
A Kondo lattice Hamiltonian for arbitrary total angular momentum J is formulated using a pseudofermion representation and without addition of RKKY interaction terms. An Hartree-Fock treatment is applied, and both variational and Green's function methods are used to calculate physical quantities from the linearized Hamiltonian. The Kondo phase is represented by finite hybridization. Magnetic ordering is examined via ordering vectors, but coexistence with the Kondo phase is not allowed. Phase diagrams are produced in S=1/2 and J=3/2 with second-order transitions at Kondo-paramagnetic and magnetic-paramagnetic boundaries, and first order transitions between Kondo and magnetic phases. Various coupling strengths are explored. Magnetic phases found include antiferromagnetism, ferromagnetism, and spin-density wave ordering of both commensurate and incommensurate varieties. In S=1/2, the magnetic phase exhibits a spike in critical temperature at half-filling. In J=3/2, the Kondo phase is reentrant at weaker coupling but not at stronger coupling.
Temple University--Theses

Lorentz TEM observations of magnetic nanoparticles contain information on the magnetic and electrostatic potentials of the sample. These potentials can be extracted from the electron wave phase shift by separating electrostatic and magnetic phase shifts, followed by 3D tomographic reconstructions. In past, Vector Field Electron Tomography (VFET) was utilized to perform the reconstruction. However, VFET is based on a conventional tomography method called filtered back-projection (FBP). Consequently, the VFET approach tends to produce inconsistencies that are prominent along the edges of the sample. We propose a model-based iterative reconstruction (MBIR) approach to improve the reconstruction of magnetic vector potential, A(r). In the case of scalar tomography, the MBIR method is known to yield better reconstructions than the conventional FBP approach, due to the fact that MBIR can incorporate prior knowledge about the system to be reconstructed. For the same reason, we seek to use the MBIR approach to optimize vector field tomographic reconstructions via incorporation of prior knowledge. We combine a forward model for image formation in TEM experiments with a prior model to formulate the tomographic problem as a maximum a posteriori probability estimation problem (MAP). The MAP cost function is minimized iteratively to deduce the vector potential. A detailed study of reconstructions from simulated as well as experimental data sets is provided to establish the superiority of the MBIR approach over the VFET approach.

L'objectiu de la tesi és estudiar la cinètica de les cristal·litzacions primàries en vidres metàl·lics mitjançant simulacions de tipus phase field. Una cristal·lització primària és una transició de fase sòlid-sòlid on la fase que cristal·litza (fase transformada o fase secundaria) té una composició química diferent de la fase precursora (fase no transformada o fase primària).
Les dades experimentals obtingudes a partir de l'estudi calorimètric de cristal·litzacions primàries s'analitzen generalment en el marc del model KJMA (Kolmogorov, Johnson & Mehl, Avrami). Aquest model proporciona l'evolució temporal de la fracció transformada basant-se en tres hipòtesis:
- Els nuclis de la fase secundaria estan distribuïts aleatòriament en tot l'espai.
- El creixement d'aquests nuclis és isotròpic.
- El creixement s'atura únicament per xoc directe (hard impingement).

En la cristal·lizació de vidres metàl·lics s'ha observat experimentalment un alentiment de la cinètica respecte del comportament calculat emprant la citada cinètica KJMA. Aquest alentiment s'explica a la literatura en base a que en aquest tipus de transformacions, controlades per difusió, la interacció entre els cristalls no és directa sinó que es produeix a través dels perfils de concentració (soft impingement) i, a més, l'evolució d'aquests perfils de concentració causa canvis en la concentració de la matriu amorfa, estabilitzant la i per tant fent que la nucleació de nous cristalls esdevingui no aleatòria. Diversos autors han proposat modificacions del model KJMA per tal d'intentar superar aquestes limitacions, basats bé en consideracions geomètriques, bé en aproximacions de camp mitjà. A pesar de tot, cap d'aquests models és capaç d'explicar satisfactòriament la cinètica observada en cristal·litzacions primàries. L'objectiu d'aquest treball ha estat la simulació realista de la cinètica de les transformacions primàries per trobar una explicació consistent a les diferències observades entre les dades experimentals i els models teòrics disponibles.
Per tal de poder descriure de forma realista el procés de cristal·lització primària s'ha d'estudiar el procés de nucleació i creixement de la fase secundaria alhora que es resol l'equació de difusió en la fase primària. En aquest treball s'ha emprat un model de simulació phase field que permet estudiar aquest sistema introduint una nova variable lligada al camp de concentració que pren dos valors diferents segons es tracti de fase transformada o no transformada. Amb aquest tipus de models també es poden introduir diferents protocols de nucleació i per tant estudiar independentment els efectes de la nucleació en la cinètica. D'aquesta manera s'han realitzat simulacions en 2 i 3 dimensions de cristal·litzacions primàries amb diferents graus de fracció transformada final). Els resultats de les simulacions s'ha comparat amb el model KJMA i, contra el que es preveia, s'ha obtingut un bon acord entre les fraccions transformades del model KJMA i de les simulacions. Donat que el model KJMA no reprodueix satisfactòriament el comportament experimental d'aquest resultat es dedueix que ni el soft impingement ni la nucleació no aleatòria son les responsables de l'alentiment de la cinètica obtingut en cristal·litzacions primàries.
Per tal de trobar una explicació físicament convincent del comportament observat experimentalment s'ha aprofundit en l'estudi teòric de les cristali·litzaciones primàries, incloent-hi l'efecte dels canvis composicionals que tenen lloc en la matriu a mesura que la transformació es produeix. Aquest fet, tot i ser conegut a la bibliografia, ha estat sistemàticament ignorat en l'elaboració de models cinètics. En concret, s'ha fet palès que canvis en la composició química de la fase primària han d'afectar de forma radical a la viscositat, que varia fortament a prop de la transició vitrea, i han de produir canvis en les propietats de transport atòmic. Això s'ha modelat a través de l'assumpció d'un coeficient de difusió depenent de la concentració, en base a la relació modificada d'Stokes-Einstein entre la viscositat i el coeficient de difusió. Les simulacions phase-field amb un coeficient de difusió d'aquest tipus donen lloc a una cinètica més lenta i que mostra un acord excel·lent amb la cinètica experimentalment observada en cristal·litzacions primàries de vidres metàl·lics. Per tant, les simulacions phase field confirmen que la cinètica de les cristal·litzacions primàries està controlada fonamentalment pel canvi en les propietats de transport atòmic, mentre que els efectes de soft impingement i nucleació no aleatoria, tot i estar presents, son secundaris.
El objetivo de la tesi es estudiar la cinética de las cristalizaciones primarias en vidrios metálicos mediante simulaciones de tipo phase field. Una cristalización primaria es una transición de fase sólido-sólido donde la fase que cristaliza (fase transformada o fase secundaria) tiene una composición química diferente a la fase precursora (fase no transformada o fase primaria).
Los datos experimentales obtenidos a partir del estudio calorimétrico de cristalizaciones primarias se analizan generalmente en el marco del modelo KJMA (Kolmogorov, Johnson & Mehl, Avrami). Este modelo proporciona la evolución temporal de la fracción transformada basándose en tres hipótesis:
- Los núcleos de la fase secundaria están distribuidos aleatoriamente en todo el espacio
- El crecimiento de estos núcleos es isotrópico
- El crecimiento se detiene únicamente por choque directo (hard impingement).

En la cristalización de vidrios metálicos se ha observado experimentalmente un retardo de la cinética respecto del comportamiento calculado usando la cinética KJMA. Este retardo se explica en la literatura en base a que en este tipo de transformaciones, controladas por difusión, la interacción entre los cristales no es directa sino que se produce a través de los perfiles de concentración (soft impingement) y, además, la evolución de estos perfiles de concentración causa cambios en la concentración de la matriz amorfa, estabilizándola y por tanto haciendo que la nucleación de nuevos cristales sea no aleatoria. Varios autores han propuesto modificaciones del modelo KJMA para intentar superar estas limitaciones, basados bien en consideraciones geométricas, bien en aproximaciones de campo medio. A pesar de todo, ninguno de estos modelos es capaz de explicar satisfactoriamente la cinética observada en cristalizaciones primarias. El objetivo de este trabajo ha sido la simulación realista de la cinética de las transformaciones primarias para hallar una explicación consistente a las diferencias entre los datos experimentales y los modelos teóricos disponibles.
Para describir de manera realista el proceso de cristalización primaria se tiene que estudiar el proceso de nucleación y crecimiento de la fase secundaria a la vez que se resuelve la ecuación de difusión en la fase primaria. En este trabajo se ha usado un modelo de simulación phase-field que permite estudiar este sistema introduciendo una nueva variable ligada al campo de concentración que toma dos valores diferentes según se trate de fase transformada o no transformada. Con este tipo de modelos también se pueden introducir diferentes protocolos de nucleación y por tanto estudiar independientemente los efectos de la nucleación en la cinética. De esta manera se han realizado simulaciones en 2 y 3 dimensiones de cristalizaciones primarias con diferentes grados de fracción transformada final. Los resultados de la simulaciones se han comparado con el modelo KJMA y, en contra de lo que se preveía, se ha obtenido un buen acuerdo entre las fracciones transformadas del modelo KJMA y de las simulaciones. Dado que el modelo KJMA no reproduce satisfactoriamente el comportamiento experimental, de este resultado se deduce que ni el soft impingement ni la nucleación no aleatoria son las responsables del retardo en la cinética obtenido en cristalizaciones primarias.
Para encontrar una explicación físicamente convincente del comportamiento observado experimentalmente se ha profundizado en el estudio teórico de las cristalizaciones primarias, incluyendo el efecto de los cambios composicionales que tienen lugar en la matriz a medida que la transformación se produce. Este hecho, aún y ser conocido en la bibliografía, ha sido sistemáticamente ignorado en la elaboración de modelos cinéticos. En concreto, se ha hecho patente que cambios en la composición química de la fase primaria tienen que afectar de forma radical a la viscosidad, que varía fuertemente cerca de la transición vítrea, y tienen que producirse cambios en las propiedades de transporte atómico. Esto se ha modelado a través de la asunción de un coeficiente de difusión dependiente de la concentración, en base a la relación de Stokes-Einstein modificada entre la viscosidad y el coeficiente de difusión. Las simulaciones phsae-field con un coeficiente de difusión de este tipo dan lugar a una cinética más lenta y que muestra un acuerdo excelente con la cinética experimentalmente observada en cristalizaciones primarias de vidrios metálicos. Por tanto, las simulaciones phase-field confirman que la cinética de las cristalizaciones primarias está controlada fundamentalmente por los cambios en las propiedades de transporte atómico, mientras que los efectos de soft-impingement y nucleación no aleatoria, aún y estar presentes, son secundarios.
The aim of this thesis is to study the kinetics of primary crystallization in metallic glasses by means of phase-field simulations. A primary crystallization is a solid-solid phase transformation where the crystallized phase (transformed phase or secondary phase) has a chemical composition different than the precursor phase (untransformed phase or primary phase).
Experimental data from calorimetric studies of primary crystallization are usually studied in the framework of the KJMA model (Kolmogorov, Johnson & Mehl, Avrami). This model yields the temporal evolution of the transformed fraction on the basis of three main assumptions:
- A random distribution of particle nuclei of the secondary phase
- The growth of these nuclei is isotropic
- The growth is only halted by direct collisions (hard impingement).

In the crystallization of metallic glasses, a slowing down of the kinetics respect the behavior calculated with the KJMA kinetics has been observed. This delay is explained in the literature by the fact that in this kind of transformations, that are diffusion controlled, the interaction between the crystals is not direct but through the concentration profiles (soft impingement) and moreover, the evolution of these profiles causes changes in the concentration of the amorphous matrix, stabilizing it and thus, the nucleation of new nuclei become non random. Several authors had proposed modifications to the KJMA model to try to overcome these limitations, based either on geometrical considerations or in mean field approaches. However, none of these models is able to explain the observed kinetics in primary crystallizations. The aim of this work has been the realistic simulation of the kinetics of primary crystallization to find a explanation to the differences between the experimental data and the available theoretical models.
In order to describe in a realistic way the process of a primary crystallization, the nucleation and growth process of the secondary phase has to be studied at the same time that the diffusion equation is solved in the primary phase. In this work, it has been used a phase field model for the simulations that allows to study this system introducing a new variable, coupled to the concentration field, that takes two different values in each of the existing phases. With these kinds of models, different nucleation protocols can also be introduced and thus, independently study the effects of the nucleation in the kinetics. Therefore, 2 and 3 dimensional simulations of primary crystallization have been performed with several degrees of final transformed fraction. The simulation results have been compared with the KJMA model and, unexpectedly, a good agreement between the simulations and the KJMA model has been obtained. As the KJMA model does not reproduce satisfactorily the experimental behavior, from this result can be deduced that neither the soft impingement nor the non random nucleation are the responsible of the slowing down observed in the kinetics of primary crystallization.
In order to find a physical convincing explanation of the observed experimental behavior, the theoretical study of primary crystallization has been extended, including the effects of the compositional changes that take place in the matrix as the transformation proceed. This fact, notwithstanding being known in the literature, has been systematically ignored in the development of the kinetics models. In particular, it has become clear that changes in the chemical composition of the primary phase have to radically affect the viscosity, that strongly varies near the glass transition, and some changes in the atomic transport properties must occur. This has been modeled through the assumption of a compositional dependent diffusion coefficient, on the basis of a modified Stokes-Einstein relation between viscosity and diffusion coefficient. Phase field simulations with a diffusion coefficient of this type yield a slower kinetics and show an excellent agreement with the kinetics experimentally observed in primary crystallization of metallic glasses. Thus, phase field simulations confirm that the kinetics of primary crystallization is fundamentally controlled by the changes in the atomic transport properties, while the soft impingement and non random effects, although being present, are secondary.

In this thesis we provide a scheme for phase separation by accounting for diffusion, dynamic equations and consistency with thermodynamics. The constituents are two compressible fluids and, for the non-simple character of the mixture, an extra energy flux is allowed to occur. Since also thermal effects are included, the result is a whole set of evolution equations for the concentration, the velocity and the temperature which describes a non-isothermal Navier-Stokes-Cahn-Hilliard model for phase separation in a binary mixture with extra fluxes and within the Fourier heat theory. Alternative heat theories may be proposed for this Navier-Stokes-Cahn-Hilliard theory. Meanwhile the mixing problem is described graphically. Moreover the model may be generalized including a source term, and this doesn't affect the thermodynamic scheme. Then we describe and then compare two mathematical models for chemotactic processes: the pioneeristic Keller-Segel model and the hydrodynamic model by Chavanis and Sire. The first one is able to describe clusters or peaks, the second one involves inertial effects together with a friction force and leads to network patterns or filaments that are in good agreement with the experimental results. We analyze the linear stability of an infinite, stationary and homogeneous distribution of cells for determining the critical thresholds above which chemotactic collapse is allowed and cellular aggregation is reproduced. Then we discuss the differences between the two models, moreover we show the analogy between the instability criterion for biological populations and the Jeans instability criterion in an astrophysical setting. Finally we propose a different approach for the derivation of new diffuse interface models for tumour growth (with chemotaxis and active transport) based on the Cahn-Hilliard theory, combined with the (stationary) Darcy momentum equation.

Este trabalho apresenta um estudo da solidificação de metais puros utilizando o modelo de campo de fases. O modelo é utilizado para simular a solidificação com o intuito de obter a morfologia da interface sólido-líquido sob diversas condições de transferência de calor. Foram realizados testes de validação comparando as morfologias da interface sólido-líquido obtida com as morfologias apresentadas em trabalhos anteriores para os casos bi e tridimensionais. O modelo do campo de fases adotado consiste principalmente de duas equações diferenciais: uma para calcular a variável de campo de fases e outra para calcular o campo de temperaturas. As equações foram solucionadas numericamente para um oitavo do domínio devido a simetria do problema. Os cálculos do modelo indicam que um sólido esférico com um raio inicial menor que o raio crítico de nucleação refunde. Entretanto uma esfera de raio maior cresce. Quando o sólido inicial cresce em uma malha numérica relativamente grosseira, a forma do sólido desvia da forma esférica devido perturbações na interface sólido-líquido. Quando a malha é refinada, as perturbações não são detectadas; contudo, quando introduzidas artificialmente as perturbações crescem e distorcem o formato esférico.
This work presents a study of the solidification of pure metals using the phase field model. The model is used to simulate solidification in order to obtain the morphology of the solid-liquid interface under different heat transfer conditions. Validation tests were performed comparing the morphology of the solid-liquid interface with the morphologies obtained from previous works for two and three dimensional cases. The adopted phase-field model consisted mainly of two differential equations: one to calculate the field of phase variable and another for the temperature field. The equations were solved numerically in only one eighth of the domain owing to the symmetry of the problem. Model calculations show that a solid sphere with an initial radius smaller than the critical radius for nucleation shrinks, whereas a sphere with a larger radius grows. When it grows in a relatively coarse numerical mesh, the initial solid shape deviates from a sphere owing to perturbations at the solid-liquid interface. When the numerical mesh is refined, the growth of perturbations is not detected, but artificially introduced perturbations grow and distort the spherical shape.

Les mémoires à changement de phases ont basées sur la variation de résistance d’un petit volume de matériau à changement de phase, l'information binaire étant codée à travers la phase amorphe ou cristalline du matériau. Le changement de phase permettant leur programmation est induit par effet Joule sous l’application d’un courant électrique. L’alliageGe2Sb2Te5 est largement utilisé pour les mémoires à changement de phase, car il cristallise rapidement et sans changement de composition. Cependant, pour obtenir la fiabilité requise pour certaines applications à haute température, notamment dans le secteur automobile, un alliage Ge-Sb-Te enrichi en Geest utilisé par la société STMicroelectronics. La cristallisation de cet alliage s’accompagne d’une ségrégation des espèces et de la formation d’une nouvelle phase cristalline. La répartition spatiale des phases et espèces est décisive pour le bon fonctionnement du point mémoire ; il est ainsi très important de pouvoir la prédire.Les modèles de champ de phase permettent,notamment aux échelles de temps et d’espace impliquées dans l’étude des mémoires à changement de phase, le suivi d’interface entre plusieurs domaines occupés par des phases différentes. Dans ce travail de thèse, un modèle multi-champ de phase permettant de simuler l’évolution de la répartition des phases et des espèces dans ce nouvel alliage a été développé.Les paramètres du modèle ont été déterminés à partir des données disponibles sur l’alliage.Deux types de simulations ont été réalisées :d’une part, celle de la cristallisation, lors d’un recuit, d’une couche mince de matériau initialement déposé amorphe ; d’autre part, celle portant sur les changements de phase qui se produisent lors de l’application de champs de température typiques des opérations d’écriture des mémoires. La comparaison entre les résultats de simulations et expériences révèle que les caractéristiques principales des microstructures observées dans les expériences sont bien mises en évidence par le modèle
Phase change memories (PCM) exploit the variation of resistance of a small volume of phase change material: the binary information is coded through the amorphous or crystalline phase of the material. The phase change is induced by an electrical current, which heats the material by the Joule effect. Because of its fast and congruent crystallization, theGe2Sb2Te5 alloy is widely used for PCM. Nevertheless, to get a better reliability at high temperatures, which is required e.g. for automotive applications, STMicroelectronics uses a Ge-rich GeSbTe alloy. In this alloy, chemical segregation and appearance of a new crystalline phase occur during crystallization. The distribution of phases and alloy components are critical for the proper functioning of the memory cell; thus, predictive simulations would be extremely useful. Phase field models are used for tracking interfaces between areas occupied by different phases. In this work, a multi-phase field model allowing simulating the distribution of phases and species in Ge-rich GeSbTe has been developed. The parameters of the model have been determined using available data on this alloy. Two types of simulations have been carried out, firstly to describe crystallization during annealing of initially amorphous deposited thin layer; secondly to follow the evolution of phase distribution during memory operation using temperature fields that are typical for those operations. Comparisons between simulations and experiments show that they both exhibit the same features