Thesis (Ph. D.)--Ohio State University, 2007. / Title from first page of PDF file. Includes bibliographical references (p. 166-181).

Dans cette thèse on étudie les effets de taille finie au-dessus de la dimension critique supérieure d_c. Les effets de taille finie y ont longtemps été incomplètement compris, en particulier vis-à-vis de leur dépendance en fonction des conditions aux limites. La violation de la relation d’échelle dite d’hyperscaling a été l’un des aspects les plus évidents des difficultés rencontrées. Le désaccord avec le scaling usuel est dû au caractère de variable non pertinente dangereuse du terme de self-interaction dans la théorie en ϕ^4. Celle-ci était considérée comme dangereuse pour la densité d’énergie libre et les fonctions thermodynamiques associées, mais pas dans le secteur des corrélations. Récemment, un schéma nouveau de scaling a été proposé dans lequel la longueur de corrélation joue un rôle central et est également affectée par la variable non pertinente dangereuse. Ce nouveau schéma, appelé QFSS, est basé sur le fait que la longueur de corrélation exhibe au lieu du scaling usuel ξ~L un comportement en puissance de la taille finie ξ~L^ϙ. Ce pseudo-exposant critique ϙ est lié à la dimension critique supérieure et à la variable dangereuse. Au-dessous de d_c, cet exposant prend la valeur ϙ=1, mais au-dessus, il vaut ϙ=d/d_c. Le schéma QFSS est parvenu à réconcilier les exposants de champs moyen et le Finite-Size-Scaling tel que dérivé du Groupe de Renormalisation pour les modèles avec interactions à courte portée au-dessus de d_c en conditions aux limites périodiques. Si ϙ est un exposant universel, la validité de la théorie doit toutefois s’étendre également aux conditions de bords libres. Des tests initiaux dans de telles conditions ont mis en évidence de nouvelles difficultés: alors que le QFSS est valable au point pseudo-critique auquel les grandeurs thermodynamiques telles que la susceptibilité manifestent un pic à taille finie, au point critique on a pensé que c’était le FSS standard qui prévalait avec les exposants de champ moyen et ξ~L. On montre dans ce travail qu’il en va différemment de la situation au point critique et qu’à la place ce sont les exposants gaussiens qui s’appliquent en l’absence de variable non pertinente dangereuse. Pour mettre en évidence ce résultat, nous avons mené des simulations de modèles avec interactions à longue portée, qui peuvent être à volonté étudiés au-dessus de leur dimension critique supérieure. Nous avons aussi développé une étude des modes de Fourier qui permet de fournir des exemples de quantités non affectées par la présence de la variable non pertinente dangereuse / In this project finite-size size scaling above the upper critical dimension〖 d〗_c is investigated. Finite-size scaling there has long been poorly understood, especially its dependency on boundary conditions. The violation of the hyperscaling relation above d_c has also been one of the most visible issues. The breakdown in standard scaling is due to the dangerous irrelevant variables presented in the self-interacting term in the ϕ^4 theory, which were considered dangerous to the free energy density and associated thermodynamic functions, but not to the correlation sector. Recently, a modified finite-size scaling scheme has been proposed, which considers that the correlation length actually plays a pivotal role and is affected by dangerous variables too. This new scheme, named QFSS, considers that the correlation length, instead of having standard scaling behaviour ξ~L , scales as ξ~L^ϙ. This pseudocritical exponent is connected to the critical dimension and dangerous variables. Below d_c this exponent takes the value ϙ=1, but above the upper critical dimension it is ϙ=d/d_c. QFSS succeeded in reconciling the mean-field exponents and FSS derived from the renormalisation-group for the models with short-range interactions above d_c with periodic boundary conditions. If ϙ is an universal exponent, the validity of that theory should also hold for the free boundary conditions. Initial tests for such systems faced new problems. Whereas QFSS is valid at pseudocritical points where quantities such as the magnetic susceptibility experience a peak for finite systems, at critical points the standard FSS seemed to prevail, i.e., mean-field exponents with ξ~L. Here, we show that this last picture at critical point is not correct and instead the exponents that applied there actually arise from the Gaussian fixed-point FSS where the dangerous variables are suppressed. To achieve this aim, we study Ising models with long-range interaction, which can be tuned above〖 d〗_c, with periodic and free boundary conditions. We also include a study of the Fourier modes which can be used as an example of scaling quantities without dangerous variables

Made available in DSpace on 2015-03-03T15:15:25Z (GMT). No. of bitstreams: 1 MarceloBS_DISSERT.pdf: 2228867 bytes, checksum: 46ad012b7ecf9d333c9b9a88bbfb0411 (MD5) Previous issue date: 2010-08-12 / Conselho Nacional de Desenvolvimento Cient?fico e Tecnol?gico / The diffusive epidemic process (PED) is a nonequilibrium stochastic model which, exhibits a phase trnasition to an absorbing state. In the model, healthy (A) and sick (B) individuals diffuse on a lattice with diffusion constants DA and DB, respectively. According to a Wilson renormalization calculation, the system presents a first-order phase transition, for the case DA > DB. Several researches performed simulation works for test this is conjecture, but it was not possible to observe this first-order phase transition. The explanation given was that we needed to perform simulation to higher dimensions. In this work had the motivation to investigate the critical behavior of a diffusive epidemic propagation with L?vy interaction(PEDL), in one-dimension. The L?vy distribution has the interaction of diffusion of all sizes taking the one-dimensional system for a higher-dimensional. We try to explain this is controversy that remains unresolved, for the case DA > DB. For this work, we use the Monte Carlo Method with resuscitation. This is method is to add a sick individual in the system when the order parameter (sick density) go to zero. We apply a finite size scalling for estimates the critical point and the exponent critical =, e z, for the case DA > DB / O processo epid?mico difusivo (PED) ? um modelo estoc?stico de n?o equil?brio que se inspira no processo de contato e que exibe uma transi??o de fase para um estado absorvente. No modelo, temos indiv?duos saud?veis (A) e indiv?duos doentes (B) se difundindo numa rede unidimensional com uma difus?o constante DA e DB, respectivamente. De acordo com os c?lculos do grupo de renormaliza??o, o sistema apresentou uma transi??o de fase de primeira ordem, para o caso DA > DB. V?rios pesquisadores realizaram trabalhos de simula??o para testar esta conjectura e n?o conseguiram observar esta transi??o de primeira ordem. A explica??o dada era que precis?vamos realizar simula??o para dimens?es maiores. Por isso, neste trabalho tivemos a motiva??o de investigarmos o comportamento cr?tico de um processo de propaga??o epid?mico difusivo com intera??o de L?vy (PEDL) em uma dimens?o. A distribui??o de L?vy tem intera??o de difus?o de todos os tamanhos levando o sistema unidimensional a um sistema de dimens?es maiores. Com isso, poderemos tentar explicar esta controv?rsia que existe at? hoje, para o caso DA > DB. Para este trabalho utilizamos o M?todo de Monte Carlo com ressuscitamento. Este m?todo consiste em acrescentar um indiv?duo doente no sistema quando o par?metro de ordem (densidade de doente) vai ? zero. Aplicamos a t?cnica de an?lise de escala de tamanho finito para determinarmos com boa precis?o o ponto cr?tico e os expoentes cr?ticos ??/v, v e z, para o caso DA > DB

This thesis presents the investigation of an Euler-Lagrange framework for modeling bubble coalescence in dispersed bubbly flows. The interaction between bubbles may be caused by several mechanisms. Among them, the random motion due to turbulent fluctuations is normally of major significance. One focus of this work is to apply a bubble dispersion model for modeling turbulence-induced coalescence, occurring in a certain percentage of collision events. Large bubbles appear due to coalescence, and their disturbance to the liquid phase is not negligible in most circumstances. However, the point-mass Euler-Lagrange method requires the bubble or particle size to be much smaller than the cell size when the interphase coupling is considered. Otherwise, numerical instabilities may arise. Therefore, interpolation methods between the Euler and the Lagrange phase for finite-size bubbles that are bigger than or of the same size as numerical cells are studied. The Euler-Lagrange method describes the continuous phase on the Euler grid, and the dispersed phase is treated as Lagrange points in the simulation. Bubble motion is governed by an ordinary differential equation for the linear momentum considering different forces. The turbulent dispersion of the dispersed phase is reconstructed with the continuous random walk (CRW) model. Bubble-bubble collisions and coalescence are accounted for deterministically. The time-consuming search for potential collision partners in dense bubbly flows is accelerated by the sweep and prune algorithm, which can be utilized in arbitrary mesh types and sizes. If the interphase coupling is considered in the simulations, the spatially distributed coupling method is used for the Lagrange-to-Euler coupling. For the Euler-to-Lagrange coupling, a new approach is proposed. To evaluate the dispersion and coalescence models, one-way coupled simulations of bubbly pipe flows at low Eötvös numbers are conducted. Validation against the experiments demonstrates that the one-way coupled EL-CRW dispersion model can well reproduce the bubble distribution in a typical dense bubbly pipe flow. Good agreement of the bubble size distribution at the pipe outlet between the simulation and the experiment is obtained. Two-way coupled simulations are performed to validate the interpolation methods. A combination of coupling approaches is employed in a square bubble column reactor to examine the general validity for a large-scale bubbly flow. Combining the proposed interpolation scheme with the dispersion and bubble interaction models, the coalescence and breakage in bubbly flows are studied in a turbulent pipe flow. The predicted bubble size distribution shows a good match to the measurement. The results are independent of the mesh resolution in the studied range from point-mass simulations to finite-size situations.:Nomenclature 1 Introduction 1.1 Motivation and background for the thesis 1.2 Outline 2 Equations for modeling bubbly flows 2.1 Governing equations of the continuous phase 2.2 Governing equations of the dispersed phase 2.3 Modifications to the bubble force equations 2.3.1 One-way coupled simulations with RANS modeling 2.3.2 Two-way coupled simulations 2.4 Generation of fluctuations 2.4.1 Different approaches to dispersion modeling 2.4.2 Normalized continuous random walk model 2.4.3 Employing the mean velocity field to determine forces 3 Bubble collision, coalescence and breakup 3.1 Previous studies and requirement of the interaction modeling 3.2 Detection of collisions with the sweep and prune algorithm 3.3 Coalescence modeling 3.3.1 Condition of bubble coalescence 3.3.2 Model of Kamp et al. [2001] 3.3.3 Model of Hoppe and Breuer [2018] 3.3.4 Model of Schwarz et al. [2013] 3.3.5 Comparison of coalescence models 3.4 Breakup modeling 3.4.1 Turbulence induced breakups 3.4.2 Post-breakup treatment 4 Interpolation techniques for two-way coupled simulations 4.1 Lagrange-to-Euler coupling 4.1.1 Introduction to the spatially distributed coupling 4.1.2 Intersection plane method 4.1.3 Subcell method 4.1.4 Random points method 4.2 Euler-to-Lagrange coupling 4.2.1 Approaches for computing the undisturbed velocity 4.2.2 Coarser grid method 4.2.3 Averaging the fluid velocity in front of the bubble 4.2.4 Velocity from upstream disk 4.2.5 Gradient of the undisturbed liquid velocity 5 One-way coupled simulation of bubble dispersion and resulting interaction 5.1 Implementation and verification of the continuous random walk model 5.2 Bubble dispersion in turbulent channel flows 5.3 Bubble dispersion and interaction in turbulent pipe flows 5.3.1 Overview of studied cases 5.3.2 Results of the bubble dispersion 5.3.3 Results of the bubble coalescence 6 Two-way coupled simulation of finite-size bubbles 6.1 Flow solver and algorithm 6.2 Assessing the Lagrange-to-Euler coupling methods 6.2.1 Previous studies 6.2.2 Simulation setups for a single bubble in quiescent liquid 6.2.3 Results and discussion 6.3 Assessing the Euler-to-Lagrange coupling methods 6.3.1 Simulation of two bubbles rising inline 6.3.2 Simulation of a bubble rising in linear shear flows 6.4 Large-eddy simulation for a square bubble column 6.5 Bubble coalescence in a turbulent pipe flow 7 Conclusions and outlook Appendices A.1 Equations of turbulence models A.2 Numerical implementation of the full CRW drift term A.3 Results of bubble coalescence modeling for case B to case E A.4 Search algorithm of the upstream disk method Bibliography / Diese Arbeit stellt die Untersuchung eines Euler-Lagrange-Rahmens zur Modellierung der Blasenkoaleszenz in dispergierten Blasenströmungen vor. Die Interaktion zwischen Blasen kann durch mehrere Mechanismen verursacht werden. Unter ihnen sind die zufälligen Bewegungen aufgrund von turbulenten Fluktuationen von großer Bedeutung. Ein Schwerpunkt dieser Arbeit ist die Anwendung eines Blasendispersionsmodells zur Modellierung der turbulenzinduzierten Koaleszenz, die in einem bestimmten Prozentsatz der Kollisionsereignisse auftritt. Große Blasen entstehen durch Koaleszenz und ihre Störung der flüssigen Phase ist in den meisten Fällen nicht zu vernachlässigen. Die Punkt-Masse-Euler-Lagrange-Methode erfordert jedoch, dass die Blasengröße viel kleiner als die Zellgröße ist, wenn die Interphasenkopplung berücksichtigt wird. Andernfalls kann es zu numerischen Instabilitäten kommen. Daher werden Interpolationsmethoden zwischen den zwei Phasen untersucht. Die kontinuierliche Phase wird auf dem Euler-Gitter beschrieben und die dispergierte Phase wird als Punkte behandelt. Die Blasenbewegung wird durch eine gewöhnliche Differentialgleichung unter Berücksichtigung verschiedener Kräfte bestimmt. Die turbulente Dispersion der Blasen wird mit dem CRW-Modell (continuous random walk) rekonstruiert. Blasen-Blasen-Kollisionen werden deterministisch berücksichtigt. Die Suche nach potentiellen Kollisionspartnern wird durch den Sweep- und Prune-Algorithmus beschleunigt, der in beliebigen Gittertypen und -größen eingesetzt werden kann. Wird die Interphasenkopplung berücksichtigt, so wird für die Lagrange-zu-Euler-Kopplung die spatially distributed coupling verwendet. Für die Euler-zu-Lagrange-Kopplung wird ein neuer Ansatz vorgeschlagen. Um die Dispersions- und Koaleszenzmodelle zu bewerten, werden Einweg-gekoppelte Simulationen von blasenbeladenen Rohrströmungen bei niedriger Eötvös-Zahl durchgeführt. Die Validierung zeigt, dass das einseitig gekoppelte EL-CRW-Dispersionsmodell die Blasenverteilung in einer typischen dichten, blasenbeladenen Rohrströmung gut reproduzieren kann. Es wird eine gute Übereinstimmung der Blasengrößenverteilung am Rohrauslass zwischen der Simulation und dem Experiment erzielt. Zur Validierung der Interpolationsmethoden werden Zweiweg-gekoppelte Simulationen durchgeführt. Eine Kombination von Kopplungsansätzen wird in einem Blasensäulenreaktor eingesetzt, um die allgemeine Gültigkeit zu untersuchen. Durch Kombination des vorgeschlagenen Interpolationsschemas mit den Dispersions- und Blasenwechselwirkungsmodellen werden die Koaleszenz und der Zerfall in einer turbulenten blasenbeladenen Rohrströmung untersucht. Die berechnete Blasengrößenverteilung zeigt eine gute Übereinstimmung mit der Messung und erweist sich als unabhängig von der Netzauflösung im untersuchten Bereich von PunktMasse-Simulationen bis zu Situationen mit Blasen endlicher Größe.:Nomenclature 1 Introduction 1.1 Motivation and background for the thesis 1.2 Outline 2 Equations for modeling bubbly flows 2.1 Governing equations of the continuous phase 2.2 Governing equations of the dispersed phase 2.3 Modifications to the bubble force equations 2.3.1 One-way coupled simulations with RANS modeling 2.3.2 Two-way coupled simulations 2.4 Generation of fluctuations 2.4.1 Different approaches to dispersion modeling 2.4.2 Normalized continuous random walk model 2.4.3 Employing the mean velocity field to determine forces 3 Bubble collision, coalescence and breakup 3.1 Previous studies and requirement of the interaction modeling 3.2 Detection of collisions with the sweep and prune algorithm 3.3 Coalescence modeling 3.3.1 Condition of bubble coalescence 3.3.2 Model of Kamp et al. [2001] 3.3.3 Model of Hoppe and Breuer [2018] 3.3.4 Model of Schwarz et al. [2013] 3.3.5 Comparison of coalescence models 3.4 Breakup modeling 3.4.1 Turbulence induced breakups 3.4.2 Post-breakup treatment 4 Interpolation techniques for two-way coupled simulations 4.1 Lagrange-to-Euler coupling 4.1.1 Introduction to the spatially distributed coupling 4.1.2 Intersection plane method 4.1.3 Subcell method 4.1.4 Random points method 4.2 Euler-to-Lagrange coupling 4.2.1 Approaches for computing the undisturbed velocity 4.2.2 Coarser grid method 4.2.3 Averaging the fluid velocity in front of the bubble 4.2.4 Velocity from upstream disk 4.2.5 Gradient of the undisturbed liquid velocity 5 One-way coupled simulation of bubble dispersion and resulting interaction 5.1 Implementation and verification of the continuous random walk model 5.2 Bubble dispersion in turbulent channel flows 5.3 Bubble dispersion and interaction in turbulent pipe flows 5.3.1 Overview of studied cases 5.3.2 Results of the bubble dispersion 5.3.3 Results of the bubble coalescence 6 Two-way coupled simulation of finite-size bubbles 6.1 Flow solver and algorithm 6.2 Assessing the Lagrange-to-Euler coupling methods 6.2.1 Previous studies 6.2.2 Simulation setups for a single bubble in quiescent liquid 6.2.3 Results and discussion 6.3 Assessing the Euler-to-Lagrange coupling methods 6.3.1 Simulation of two bubbles rising inline 6.3.2 Simulation of a bubble rising in linear shear flows 6.4 Large-eddy simulation for a square bubble column 6.5 Bubble coalescence in a turbulent pipe flow 7 Conclusions and outlook Appendices A.1 Equations of turbulence models A.2 Numerical implementation of the full CRW drift term A.3 Results of bubble coalescence modeling for case B to case E A.4 Search algorithm of the upstream disk method Bibliography

Untersucht wird der Auswirkung von Anisotropie auf den unordnungsinduzierten Metall-Isolator-Übergang (MIÜ) im Rahmen des dreidimensionalen Anderson-Modells der Lokalisierung für (schwach) gekoppelte Ebenen bzw. Ketten. Mittels numerischer Verfahren (Lanczos- und Transfer-Matrix-Methode) werden Eigenwerte und -vektoren bzw. die Lokalisierungslänge berechnet. Zur Bestimmung des kritischen Exponenten dieses Phasenüberganges 2. Ordnung wird ein allgemeiner Skalenansatz verwendet, der auch den Einfluss einer irrelevanten Skalenvariablen und Nichtlinearitäten berücksichtigt. Ein Kapitel untersucht die verwendeten numerischen Verfahren, verschiedene Methoden werden verglichen und die Portierbarkeit zu Parallelrechnern diskutiert. Der MIÜ wird mit zwei unabhängigen Methoden charakterisiert: Eigenwertstatistik und Transfer-Matrix-Methode. Die Systemgrößenunabhängigkeit der betrachteten Größen am Phasenübergang wird benutzt um den MIÜ zu identifizieren. Sie resultiert aus der Multifraktalität der kritischen Eigenzustände, die für den isotropen Fall bis zu einer Systemgröße von 111^3 Gitterplätzen gezeigt wird. Es stellt sich heraus, daß der MIÜ auch bei sehr starker Anisotropie existiert und bereits bei geringerer Potentialunordnung als im isotropen Fall auftritt. Für den Fall sehr schwach gekoppelter Ebenen wird gezeigt, daß der kritische Exponent mit dem des isotropen Falles übereinstimmt und damit die übliche Einteilung in Universalitätsklassen bestätigt.

Le modèle de boucles diluées \(A_2^{(2)}\) est étudié pour la géométrie d'un ruban de taille \(N\). Deux familles de conditions frontières sont connues pour satisfaire l’équation de Yang-Baxter à la frontière. Fixer ces conditions aux deux bouts du ruban donne un total de quatre modèles. Pour chaque modèle, les matrices de transfert, qui commutent entre elles, sont connues. Dans ce mémoire, la hiérarchie de fusion des matrices de transfert et les systèmes T et Y sont construits pour chaque modèle et pour un paramètre de croisement \(\lambda\) générique. Pour \(\lambda/\pi\) rationnel, il est prouvé qu'il existe une relation linéaire entre les matrices fusionnées qui ferme la hiérarchie de fusion en un système fini. Les relations de fusion et de fermeture permettent de calculer les premiers termes d'une expansion de l'énergie libre lorsque \(N\) est grand. Ces premiers termes correspondent à l'énergie libre de bulk et de bord. Les résultats analytiques sont en accord avec des résultats numériques obtenus pour de petits \(N\). Ce mémoire complète une étude des modèles \(A_2^{(2)}\) avec des conditions frontières périodiques (A. Morin-Duchesne, P.A. Pearce, J. Stat. Mech. (2019)). / We study the dilute \(A_2^{(2)}\) loop models on the geometry of a strip of width \(N\). Two families of boundary conditions are known to satisfy the boundary Yang-Baxter equation. Fixing the boundary condition on the two ends of the strip leads to four models. We construct the fusion hierarchy of commuting transfer matrices for the model as well as its T- and Y-systems, for these four boundary conditions and with a generic crossing parameter \(\lambda\). For \(\lambda/\pi\) rational we prove a linear relation satisfied by the fused transfer matrices that closes the fusion hierarchy into a finite system. The fusion relations allow us to compute the two leading terms in the large-\(N\) expansion of the free energy, namely the bulk and boundary free energies. These are found to be in agreement with numerical data obtained for small \(N\). The present work complements a previous study (A. Morin-Duchesne, P.A. Pearce, J. Stat. Mech. (2019)) that investigated the dilute \(A_2^{(2)}\) loop models with periodic boundary conditions.

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The dissertation covers phase transitions in the realm of the Anderson model of localization on topologically disordered Voronoi-Delaunay lattices. The disorder is given by random connections which implies correlations due to the restrictive lattice construction. Strictly speaking, the system features "strong anticorrelation", which is responsible for quenched long-range fluctuations of the coordination number. This attribute leads to violations of universal behavior in various system, e.g. Ising and Potts model, and to modifications of the Harris and the Imry-Ma criteria. In general, these exceptions serve to further understanding of critical phenomena. Hence, the question arises whether such deviations also occur in the realm of the Anderson model of localization in combination with random Voronoi-Delaunay lattice. For this purpose, four cases, which are distinguished by the spatial dimension of the systems and by the presence or absence of a magnetic field, are investigated by means of two different methods, i.e the multifractal analysis and the recursive Green function approach. The behavior is classified by the existence and type of occurring phase transitions and by the critical exponent v of the localization length. The results for the four cases can be summarized as follows. In two-dimensional systems, no phase transitions occur without a magnetic field, and all states are localized as a result of topological disorder. The behavior changes under the influence of the magnetic field. There are so-called quantum Hall transitions, which are phase changes between two localized regions. For low magnetic field strengths, the resulting exponent v ≈ 2.6 coincides with established values in literature. For higher strengths, an increased value, v ≈ 2.9, was determined. The deviations are probably caused by so-called Landau level coupling, where electrons scatter between different Landau levels. In contrast, the principle behavior in three-dimensional systems is equal in both cases. Two localization-delocalization transitions occur in each system. For these transitions the exponents v ≈ 1.58 and v ≈ 1.45 were determined for systems in absence and in presence of a magnetic field, respectively. This behavior and the obtained values agree with known results, and thus no deviation from the universal behavior can be observed.:1. Introduction 2. Random Voronoi-Delaunay lattice 2.1. Definition 2.2. Properties 2.3. Numerical construction 3. Anderson localization 3.1. Conventional Anderson transition 3.1.1. Fundamentals 3.1.2. Scaling theory of localization 3.1.3. Universality 3.2. Quantum Hall transition 3.2.1. Universality 3.3. Random Voronoi-Delaunay Hamiltonian 4. Methods 4.1. Multifractal analysis 4.1.1. Fundamentals 4.1.2. Box-size scaling 4.1.3. Partitioning scheme 4.1.4. Numerical realization 4.2. Recursive Green function approach 4.2.1. Fundamentals 4.2.2. Recursive formulation 4.2.3. Layer construction 4.3. Finite-size scaling approach 4.3.1. Scaling functions 4.3.2. Numerical determination 5. Electron behavior on 2D random Voronoi-Delaunay lattices 5.1. 2D orthogonal systems 5.2. 2D unitary systems 5.2.1. Density of states and principal behavior 5.2.2. Criticality in the lowest Landau band 5.2.3. Criticality in higher Landau bands 5.2.4. Edge states 6. Electron behavior on 3D random Voronoi-Delaunay lattices 6.1. 3D orthogonal systems 6.1.1. Pure connectivity disorder 6.1.2. Additional potential disorder 6.2. 3D unitary systems 6.2.1. Pure topological disorder 7. Conclusion Bibliography A. Appendices A.1. Quantum Hall effect on regular lattices A.1.1. Simple square lattice A.1.2. Triangular lattice A.2. Further quantum Hall transitions on 2D random Voronoi-Delaunay lattices Lebenslauf Publications / Diese Dissertation behandelt Phasenübergange im Rahmen des Anderson-Modells der Lokalisierung in topologisch ungeordneten Voronoi-Delaunay-Gittern. Die spezielle Art der Unordnung spiegelt sich u.a. in zufälligen Verknüpfungen wider, welche aufgrund der restriktiven Gitterkonstruktion miteinander korrelieren. Genauer gesagt zeigt das System eine "starke Antikorrelation", die dafür sorgt, dass langreichweitige Fluktuationen der Verknüpfungszahl unterdrückt werden. Diese Eigenschaft hat in anderen Systemen, z.B. im Ising- und Potts-Modell, zur Abweichung vom universellen Verhalten von Phasenübergängen geführt und bewirkt eine Modifikation von allgemeinen Aussagen, wie dem Harris- and Imry-Ma-Kriterium. Die Untersuchung solcher Ausnahmen dient zur Weiterentwicklung des Verständnisses von kritischen Phänomenen. Somit stellt sich die Frage, ob solche Abweichungen auch im Anderson-Modell der Lokalisierung unter Verwendung eines solchen Gitters auftreten. Dafür werden insgesamt vier Fälle, welche durch die Dimension des Gitters und durch die An- bzw. Abwesenheit eines magnetischen Feldes unterschieden werden, mit Hilfe zweier unterschiedlicher Methoden, d.h. der Multifraktalanalyse und der rekursiven Greensfunktionsmethode, untersucht. Das Verhalten wird anhand der Existenz und Art der Phasenübergänge und anhand des kritischen Exponenten v der Lokalisierungslänge unterschieden. Für die vier Fälle lassen sich die Ergebnisse wie folgt zusammenfassen. In zweidimensionalen Systemen treten ohne Magnetfeld keine Phasenübergänge auf und alle Zustände sind infolge der topologischen Unordnung lokalisiert. Unter Einfluss des Magnetfeldes ändert sich das Verhalten. Es kommt zur Ausformung von Landau-Bändern mit sogenannten Quanten-Hall-Übergängen, bei denen ein Phasenwechsel zwischen zwei lokalisierten Bereichen auftritt. Für geringe Magnetfeldstärken stimmen die erzielten Ergebnisse mit den bekannten Exponenten v ≈ 2.6 überein. Allerdings wurde für stärkere magnetische Felder ein höherer Wert, v ≈ 2.9, ermittelt. Die Abweichungen gehen vermutlich auf die zugleich gestiegene Unordnungsstärke zurück, welche dafür sorgt, dass Elektronen zwischen verschiedenen Landau-Bändern streuen können und so nicht das kritische Verhalten eines reinen Quanten-Hall-Überganges repräsentieren. Im Gegensatz dazu ist das Verhalten in dreidimensionalen Systemen für beide Fälle ähnlich. Es treten in jedem System zwei Phasenübergänge zwischen lokalisierten und delokalisierten Bereichen auf. Für diese Übergänge wurde der Exponent v ≈ 1.58 ohne und v ≈ 1.45 unter Einfluss eines magnetischen Feldes ermittelt. Dieses Verhalten und die jeweils ermittelten Werte stimmen mit bekannten Ergebnissen überein. Eine Abweichung vom universellen Verhalten wird somit nicht beobachtet.:1. Introduction 2. Random Voronoi-Delaunay lattice 2.1. Definition 2.2. Properties 2.3. Numerical construction 3. Anderson localization 3.1. Conventional Anderson transition 3.1.1. Fundamentals 3.1.2. Scaling theory of localization 3.1.3. Universality 3.2. Quantum Hall transition 3.2.1. Universality 3.3. Random Voronoi-Delaunay Hamiltonian 4. Methods 4.1. Multifractal analysis 4.1.1. Fundamentals 4.1.2. Box-size scaling 4.1.3. Partitioning scheme 4.1.4. Numerical realization 4.2. Recursive Green function approach 4.2.1. Fundamentals 4.2.2. Recursive formulation 4.2.3. Layer construction 4.3. Finite-size scaling approach 4.3.1. Scaling functions 4.3.2. Numerical determination 5. Electron behavior on 2D random Voronoi-Delaunay lattices 5.1. 2D orthogonal systems 5.2. 2D unitary systems 5.2.1. Density of states and principal behavior 5.2.2. Criticality in the lowest Landau band 5.2.3. Criticality in higher Landau bands 5.2.4. Edge states 6. Electron behavior on 3D random Voronoi-Delaunay lattices 6.1. 3D orthogonal systems 6.1.1. Pure connectivity disorder 6.1.2. Additional potential disorder 6.2. 3D unitary systems 6.2.1. Pure topological disorder 7. Conclusion Bibliography A. Appendices A.1. Quantum Hall effect on regular lattices A.1.1. Simple square lattice A.1.2. Triangular lattice A.2. Further quantum Hall transitions on 2D random Voronoi-Delaunay lattices Lebenslauf Publications

La compréhension des mécanismes qui pilotent la transmission des contraintes aux interfaces déformables est au centre de nombreuses problématiques touchant des applications actuelles utilisant un film mince de polymère souple comme couche interfaciale. Arriver à caractériser de tels films fins est encore un défi aujourd’hui car l’analyse des mesures expérimentales destinées à extraire les contributions dues aux films est complexe et délicate et les techniques usuelles de caractérisation sont peu adaptées aux systèmes. Ce travail étudie la réponse mécanique de deux types de systèmes modèles au moyen de deux techniques de caractérisation différentes. Le premier système que nous avons élaboré et caractérisé mécaniquement par le test JKR, est constitué de films d’élastomère réticulé d’épaisseurs micrométriques (de 5 à 100µm) et déposés sur des wafers de silicium. Les mesures expérimentales ont été analysées par comparaison à un modèle semi-analytique récent proposé par E. Barthel dans le but d’extraire le module élastique de chaque film et de répondre à la question de savoir si l’épaisseur du film influe sur la valeur de ce module. Nous avons montré que ce modèle permet de rendre compte quantitativement du raidissement lié à la présence d’un solide supportant le film mais que la précision sur les mesures de modules de Young reste limitée (de l’ordre de 35 %).Le deuxième système modèle est constitué de brosses de polymères greffées (PDMS) par une extrémité à la surface de wafers de silicium et gonflées dans un bon solvant (47V20). Nous avons analysé la réponse mécanique dans plusieurs régimes de distance et de fréquence en utilisant un appareil à forces de surface (SFA) dans lequel on contrôle l’approche d’une sphère millimétrique d’un plan sur lequel sont greffées les polymères. En statique, nous avons vérifié que la réponse en compression était celle d’une brosse de type Alexander-de Gennes. En mode dynamique, quand la sphère est loin de la couche gonflée, nous avons vérifié que la réponse dissipative était celle d’un écoulement de Reynolds qui décrit normalement l’écoulement d’un fluide simple newtonien entre une sphère et un plan solide. Ceci nous a permis de montrer que l’écoulement du solvant pénètre partiellement à l’intérieur de la couche greffée sur une profondeur de l’ordre du tiers de l’épaisseur gonflée de la couche. Dans le régime ou les brosses sont comprimées, il n’y a pas d’accord entre les mesures réalisées et le modèle classique de Fredrickson et Pincus. Ceci s’explique par les expériences que nous avons réalisées sur un substrat nu (sans polymère) montrant pour la première fois la déformation des substrats solides qui sont indentés par l’écoulement de liquide et qu’il faut prendre en compte cette déformation dans les analyses de nanorhéologie. Finalement, une annexe est consacrée à la fabrication de surfaces hydrophobes silanisées optimisées en vue d’étudier le glissement d’un liquide simple et d’électrolytes à la paroi. / Understanding how stresses are transmitted to deformable interfaces is a key-point in numerous issues having everyday life applications which use a thin polymer film as an interfacial layer. Still, characterizing the mechanical properties of such elastic films remains a challenge because the usual employed techniques are destructive of the surface and because of the complexity of the associated analysis. In this work, we study the mechanical response of two types of home-made model systems using two different characterization techniques. The first system – studied with a JKR test- is composed of reticulated elastomeric films of micrometric thickness (5 to 100 µm) and stuck to a silicon wafer. We analyse the experimental data with E.Barthel’s recently published semi-analytical model in order to determine the elastic modulus of each indented film and see if the thickness of the film had any influence on its value. We show that this model is in a quantitatively good agreement with our data but that we only have a 35% accuracy on the elastic modulus values thanks to the set-up. The second system we studied consists in polymer brushes end-grafted onto the surface of silicon wafers and of nanometric thickness. To characterize the mechanical response of those brushes and the effect of both their molecular organization and ingredients on their ability to transmit stresses at the interface, we use a surface force apparatus in the dynamic mode as a soft fluid indenter. We use a millimetric sphere to create a liquid flow of the solvent in which the brushes are immerged and swollen. This flow induces hydrodynamic forces whose range we can control by varying the excitation frequency and the distance of approach. We obtain the following results : first with the static response we checked that the response of the polymer layers are well-described by the Alexander-de Gennes approximation. In the dynamical mode, when the sphere is far from the solid surface, we showed that the dissipative response was well-described by the Reynolds force. Thanks to those results, we succeeded in localizing the limit of penetration of the liquid flow inside the brushes at one third of the thickness of the swollen brush; second, when the brushes are compressed, we showed that the existing models (Fredrickson & Pincus) are insufficient to explain the dynamic responses of the brushes. This disagreement is explained by experiments we performed on the bare solid substrate, which show for the first time, the deformation of the substrate due to the liquid. Thus, the mechanical response of the underlying substrate has to be taken into account in the analysis of the nanorheological results on the brushes even though the substrate is much stiffer than the polymer layers. Finally, we present how we fabricated hydrophobic (silanized) surfaces in order to study the sliding of simple liquids at the wall with the same surface force apparatus.

O modelo raise and peel é um modelo estocástico unidimensional com absorção local e desorção não local. O modelo depende de um único parâmetro u que é a razão entre a taxa de absorção pela de dessorção. Em um valor especial deste parâmetro (u = 1) o modelo tem características interessantes. O espectro é descrito por uma teoria de campos conforme (carga central c = 0), sendo que a distribuição de probabilidade estacionária está relacionada a um sistema de equilíbrio em duas dimensões. O diagrama de fases do modelo, como função do parâmetro u, tem uma fase massiva (com lacuna de massa) e uma sem massa (lacuna de massa nula) com expoentes críticos que variam continuamente com o parâmetro u. Nesta dissertação estudamos uma extensão do modelo raise and peel model no ponto u = 1, e que depende de um parâmetro adicional p. Surpreendentemente o novo modelo exibe invariância conforme para todo o domínio do seu parâmetro p, e está na mesma classe de universalidade do modelo raise and peel usual (u = 1). A única diferença entre os dois modelos é o valor da velocidade do som vs(p), que agora é função de p. Os métodos que utilizamos nesta dissertação foram diagonalizações exatas do operador de evolução do modelo (Hamiltoniano) para cadeias pequenas e simulações de Monte Carlo. / The raise and peel model is a one-dimensional nonlocal stochastic model where adsorption happens locally and desorption is nonlocal. The model depends on the single parameter u that is the ratio among the desorption and adsorption rates. At a special value of this parameter (u = 1) the model has interesting features. The spectrum is described by a conformal field theory (central charge c = 0), and its stationary probability density is related to the equilibrium distribution of a two dimensional system. The phase diagram of the model, as a function of the parameter u, has a massive phase (gapped phase) and a massless (gapless phase) whose critical exponents vary continuously with u. In this monography we study a one-parameter extension of the raise and peel model at u = 1, that depends on the additional parameter p. The new model exhibits conformal invariance for the whole range of values of its parameter p, and it is in the same universality class as the usual raise and peel model. The single difference between the models is the value of the sound velocity vs(p) which is a function of p. The methods used in this monography are the exact diagonalization of the evolution operator of the stochastic model (Hamiltonian), for small lattice sizes and Monte Carlo simulations.