Matter exists in many different phases, for example in solid state or in liquid phase. There are also phases in which the ordering of atoms is the same, but which differ in some other respect, for example ferromagnetic and paramagnetic states. According to Landau's symmetry breaking theory every phase transition is connected to a symmetry breaking process. A solid material has discrete translational symmetry, while liquid phase has continuous translational symmetry. But it has turned out that there also exist phase transitions that can occur without a symmetry breaking. This phenomenon is called topological order. In this thesis we consider one example of a theoretical model constructed on a two dimensional lattice in which one obtains topological order.

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In dieser Arbeit wird ein Modell für das Eu-reiche EuO formuliert. Es besteht in einer Erweiterung des Kondo-Gitter-Modells (KGM). Für das KGM existieren nur einige exakte Aussagen. In dieser Arbeit kommt eine neue hinzu, nämlich die exakte Abbildung des periodischen Anderson-Modells auf das antiferromagnetische KGM für beliebige Kopplungsstärke J. Reines EuO ist ein ferromagnetischer Halbleiter. Eu-reiches EuO zeigt einen gewaltigen Metall-Isolator-Übergang in der Nähe der Curie-Temperatur mit einem Sprung im Widerstand von bis zu 13 Größenordnungen. Das ist der größte Sprung im Widerstand, der jemals in der Natur beobachtet wurde. Wir reproduzieren diesen Sprung theoretisch mit der Kubo-Formel. Wir erzielen sehr gute Fits bereits in einer nicht vollständig selbstkonsistenten Theorie, bei der die Magnetisierung der Eu-Spins einer Brillouin-Funktion entnommen ist. In einer vollständig selbstkonsistenten Theorie bestimmen wir die Magnetisierung, die Curie-Temperatur, den spezifischen Widerstand und andere Transporteigenschaften. Wir berechnen Größen wie die elektronische Wärmeleitfähigkeit und die Thermokraft, für die weniger experimentelle Daten zum Vergleich vorhanden sind. Nichtsdestoweniger erscheinen z.B. die Rechnungen für die thermische Leitfähigkeit vertrauenswürdig, da das Wiedemann-Franz-Verhältnis mit der elektrischen Leitfähigkeit einen vernünftigen Wert liefert. Die Leitungselektronenzahl des Eu-reichen EuO kommt aus der Theorie unabhängig von der Leitfähigkeit heraus. Daher können wir aus der Leitfähigkeit und der Leitungselektronenzahl die durchschnittliche Drude-Mobilität (oder Streuzeit) berechnen. Diese Größe hat für höhere Impurity-(Sauerstoff-Leerstellen)-Konzentrationen einen Sprung in der Nähe der Curie-Temperatur von bis zu zwei Größenordnungen in Übereinstimmung mit dem Experiment. / In this thesis a model is formulated for Eu-rich EuO. It consists in an extension of the Kondo lattice model (KLM). For the KLM only a few exact statements exist. To those we add a new one, namely the exact mapping of the periodic Anderson model on the antiferromagnetic KLM for arbitrary coupling constant J. Pure EuO is a ferromagnetic semiconductor. Eu-rich EuO exhibits a huge metal--insulator transition near the Curie temperature with a jump in resistivity of up to 13 orders of magnitude. It is the biggest jump in resistivity ever observed in nature. We theoretically reproduce this jump. We achieve very good fits already within a not fully self-consistent theory where the magnetization of the Eu spins is taken from a Brillouin function. In a fully self-consistent theory we determine the magnetization, the Curie temperature, the resistivity and other transport properties. We calculate quantities like the electronic thermal conductivity and the thermopower, for which there are less experimental data to compare with. Nevertheless, e.g. the calculations for the thermal conductivity seem reliable since the Wiedemann-Franz ratio with the electrical conductivity gives a reasonable result. The conduction-electron number of Eu-rich EuO comes out of the theory independently of the conductivity. So we can calculate from the conductivity and the conduction-electron number the average Drude mobility (or scattering time). This quantitiy has a jump near the Curie temperature of up to two orders of magnitude for higher impurity (oxygen vacancy) concentrations in agreement with the experiment.

Die vorliegende Arbeit befasst sich mit dem ferromagnetischen Kondo-Gitter-Modell (s-d-, s-f-Modell) für Filmstrukturen. Die Spin-Fermion-Wechselwirkung des Modells kommt in Materialien vor, in denen lokalisierte Spins mit beweglichen Ladungsträgern wechselwirken, wie etwa in (verdünnten) magnetischen Halbleitern, Manganaten, oder Seltene-Erd-Verbindungen. Die durch die Ladungsträger vermittelte, indirekte Wechselwirkung zwischen den lokalisierten Spins reicht von der langreichweitigen, oszillierenden RKKY-Austauschwechselwirkung im Falle schwacher Kopplung bis zur kurzreichweitigen Doppelaustausch-Wechselwirkung bei starker Spin-Fermion-Kopplung. Beide Grenzfälle werden in dieser Arbeit durch die Abbildung des Problems auf ein effektives Heisenberg-Modell erfasst. Der Einfluss von reduzierter Translationssymmetrie auf die effektive Austauschwechselwirkung und auf die magnetischen Eigenschaften des ferromagnetischen Kondo-Gitter-Modells wird untersucht. Curie-Temperaturen werden für verschiedene Parameterkonstellationen berechnet. Die Auswirkungen von Ladungstransfer und von Gitter-Relaxation auf die magnetische Oberflächenstabilität werden betrachtet. Die Diskussion bezieht sich auf die Modifizierungen der Zustandsdichte und der kinetischen Energie im dimensionsreduzierten Fall, da die effektiven Austauschintegrale eng mit diesen Größen verknüpft sind. Die Bedeutung von Spinwellen für den Magnetismus dünner Filme und an der Oberfläche wird gezeigt. Die Interlagen-Austauschkopplung stellt ein besonders interessantes und wichtiges Beispiel der indirekten Wechselwirkung zwischen lokalisierten Momenten dar. Im Rahmen einer RKKY-Behandlung wird die Kopplung zwischen Monolagen in dünnen Filmen untersucht. Sie wird entscheidend durch die Art der ebenen und senkrechten Ladungsträgerdispersion bestimmt und ist jenseits eines kritischen Wertes der Fermi-Energie stark unterdrückt. Schließlich wird die temperaturabhängige magnetische Stabilität von interlagen-gekoppelten dünnen Filmen behandelt und die Bedingungen für einen temperaturgetriebenen magnetischen Reorientierungsübergang werden diskutiert. / This thesis is concerned with the ferromagnetic Kondo lattice (s-d, s-f) model for film geometry. The spin-fermion interaction of this model refers to materials in which localized spins interact with mobile charge carriers like in (dilute) magnetic semiconductors, manganites, or rare-earth compounds. The carrier-mediated, indirect interaction between the localized spins comprises the long-range, oscillatory RKKY exchange interaction in the weak-coupling case and the short-range double-exchange interaction for strong spin-fermion coupling. Both limits are recovered in this work by mapping the problem onto an effective Heisenberg model. The influence of reduced translational symmetry on the effective exchange interaction and on the magnetic properties of the ferromagnetic Kondo lattice model is investigated. Curie temperatures are obtained for different parameter constellations. The consequences of charge transfer and of lattice relaxation on the magnetic stability at the surface are considered. Since the effective exchange integrals are closely related to the electronic structure in terms of the density of states and of the kinetic energy, the discussion is based on the modifications of these quantities in the dimensionally-reduced case. The important role of spin waves for thin film and surface magnetism is demonstrated. Interlayer exchange coupling represents a particularly interesting and important manifestation of the indirect interaction among localized magnetic moments. The coupling between monatomic layers in thin films is studied in the framework of an RKKY approach. It is decisively determined by the type of in-plane and perpendicular dispersion of the charge carriers and is strongly suppressed above a critical value of the Fermi energy. Finally, the temperature-dependent magnetic stability of thin interlayer-coupled films is addressed and the conditions for a temperature-driven magnetic reorientation transition are discussed.

Made available in DSpace on 2014-06-11T19:30:54Z (GMT). No. of bitstreams: 0 Previous issue date: 2008-04-28Bitstream added on 2014-06-13T18:41:06Z : No. of bitstreams: 1 oliveira_lc_dr_sjrp.pdf: 1552331 bytes, checksum: ff9be69b536d540081c28a83bf038868 (MD5) / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / A topologia do estado nativo de uma proteína desempenha um papel crucial no processo de enovelamento. Neste trabalho uma nova aproximação utilizando aspectos topol ogicos para investigar a evolução protéica e apresentada. O modelo utiliza uma rede c ubica 3 3 3 de 27 monômeros e um mapa de conexões entre diferentes conformações em espa co de fase estrutural e de sequência. Desenhamos a melhor sequência não frustrada para cada uma das 103346 conformações maximamente compactas usando um algorítimo que maximiza o número de tipos de monômeros na sequência. Isto significa que cada sequência não pode possuir contatos desfavor aveis. O n umero m aximo de tipos de monômeros e 5. A sequência-conformação e considerada \protein-like se ela tem uma unica conformação de mais baixa energia, alem de acessibilidade e robustez. De todas as conformações maximamente compactas, somente 4; 75% geraram sequências \protein-like', o qual são o alvo neste estudo. Com esses dados realizamos simulações de Monte Carlo (MC) no qual examinamos as melhores sequêencias estruturas baseando-se no ZScore. A simulação e iniciada com uma sequência aleatória no qual e testada em todas as conformações, seguindo as regras estipuladas por MC. Se o ZScore aumenta, assumimos que a nova conformação e mais estável que a anterior. Esse processo e repetido até que as sequências otimamente desenhadas (com mais alto ZScore) são alcançadas. Mantendo as trajetórias originadas via MC, um mapa de conectividade sequências-estruturas e obtido. Os resultados mostram trajetórias conectadas com estruturas com baixos valores de ZScore. O aumento do ZScore ao longo da simulação conduz a um pequeno grupo de conformações preferenciais. O modelo sugere um funil de estruturas para a evolução de proteínas no qual as estruturas do fundo estão associadas com o ii \motif de uma proteína... / The topology of a protein native state plays a crucial role in the folding process. In this work a new approach using topological aspects to investigate the protein evolutions is presented. The model uses the 27-mer in a cubic lattice of 3 3 3, and a conection map between di erent conformations is found in the sequence and structural phase space. We designed the best unfrustrated sequence for each of the 103346 maximally compact conformation, using an algorithm that maximizes the number for monomers types in the sequence. This means that each sequence cannot have unfavorable contacts. The maximum number of types of monomer is 5. The sequence-conformation is considered protein-like if it has a unique lowest energy conformation, accessible and robust . Out of all maximally compact conformations, only 4,75% generated protein-like sequence, with are targeted in this study. With this data we performed a Monte Carlo simulations in which we probe for better sequence-structure based on Zscore. The simulation start which a random sequence and it is tested all conformations, nding its conformations according to the Monte Carlo rules. If the Zscore increases, we assume that the new conformation is more stable than the previous. This process is repeated until the optimally designed sequence (with the highest Zscore) is reached. Keeping track of all the Monte Carlo trajectories, a map of conectivity of sequence-structures is obtained. The results shows trajectories connected with structures of low Zscore values. The increase of Zscore along of the simulation leads to a small group of preferred conformations. The model suggest a funnel like structure for folding evolution, in which the structures at the bottom of the funnel are associated with the motif of a protein. This result can be a possible iv explanation for the restricted number of conformations compared to the large number of sequences... (Complete abstract click electronic access below)

In dieser Arbeit wird das eindimensionale Kondo-Gitter-Modell untersucht, das die Wechselwirkung eines Gitters lokaler magnetischer Momente mit unkorrelierten Leitungselektronen beschreibt. Mit Hilfe der Methode der kontinuierlichen unitären Transformationen (Flussgleichungen) wird das Modell im Parameterbereich schwacher Wechselwirkungsstärke betrachtet. In diesem Bereich zeigt das Modell so genanntes Luttinger-Flüssigkeitsverhalten. Im Rahmen der Flussgleichungsmethode wird der Hamilton-Operator auf ein effektives Modell abgebildet, in dem Elektronen und Spinmomente vollständig entkoppelt sind. Das Resultat dieses Prozesses ist ein Modell, das ein nichtwechselwirkendes Elektronengas und eine Heisenberg-Spinkette beschreibt. Das Eigenwertproblem der Heisenberg-Kette wird im Rahmen einer Schwinger-Boson-Molekularfeld-Theorie beschrieben. Zur Charakterisierung der Grundzustandseigenschaften des eindimensionalen Kondo-Gitter-Modells wurden verschiedene Erwartungswerte und Korrelationsfunktionen betrachtet. Neben statischen Größen, wie der Ladungskorrelationsfunktion der Elektronen oder der Spinkorrelationsfunktion der lokalen Spinmomente, werden dynamische Größen, wie die elektronische Zustandsdichte oder die dynamischen Spinstrukturfaktoren der Elektronen und der lokalen Spinmomente, berechnet. / The one-dimensional Kondo lattice model is investigated. This model describes the interaction between a lattice of local magnetic moments and uncorrelated conduction electrons. It is studied by means of the continuous unitary transformation's method (flow equations) within the parameter regime of weak interaction strength. Here the model shows so called Luttinger liquid behaviour. Within the framework of the flow equation's method the original Hamiltonian is mapped on an effective model, where electrons and local moments are completely decoupled. The result of this process is a model describing a non-interacting electron gas and a Heisenberg spin chain. The eigenvalue problem of the Heisenberg chain is described within a Schwinger bosons molecular field theory. In order to characterise the ground state properties of the one-dimensional Kondo lattice model different expectation values and correlation functions are investigated. Beside static properties like the charge correlation function of the electrons or the local moment's spin correlation function, dynamic properties are determined, like the electronic density of states or the dynamic spin structure factor of both the electrons and the local moments.

The phase behavior of CO2 + polymer systems is of interest in polymer synthesis, flue and natural gas processing, polymer foam and nanoparticle processing, and drug delivery. Theoretical and experimental evidence suggests that CO2 is able to interact with electron donating functional groups in polymers to form weak Lewis acid – base or EDA (Electron Donor Acceptor) complexes. These complexes can have a significant effect on the phase behavior of associated CO2 + polymer systems. In spite of this, however, the phase equilibria of only a few associated CO2 + polymer systems have been measured. Some success in modeling the phase behavior of polymer solutions has been achieved by various versions of the Statistical Association Fluid Theory (SAFT), as well as by several Lattice Models. However, many of these models incorporate two to four adjustable parameters that often depend on temperature (T), pressure (P), and/or molecular weight (MW). As a result, a large amount of experimental data is required to apply these models. The goal of the present work was therefore to develop a new thermodynamic model for associating systems that would include no more than two temperature-independent adjustable parameters. The new model presented in this work is based on the Guggenheim-Huggins-Miller lattice and includes complex formation in the development of the partition function. The EOS obtained from the resulting partition function includes two mixture parameters – the enthalpy of association or complex formation and a reference value of the equilibrium constant for complex formation . Most importantly, can be obtained from in situ Attenuated Total Reflection Fourier Transform Infrared (ATR – FTIR) measurements. This work therefore demonstrates the use of ATR – FTIR spectra to obtain molecular level information regarding the interaction of CO2 and electron donating functional groups in polymers. Unlike other studies, this work uses the bending vibration of CO2 to estimate the enthalpies of association ( ) of CO2 + polymer systems. Values of were directly incorporated in the new model and were found to lie between -7 and -12 kJ/mol for the systems investigated in this work. They increased (i.e. became more negative) in the order: CO2 + PS-co-PMMA < CO2 + PMMA <CO2 + PBMA < CO2 + PSF < CO2 + PVAc < CO2 + EVA40 < CO2 + PEG. Values of the second parameter in the new EOS ( ) were obtained by fitting solubility data at one temperature. Both and were found to be temperature independent. The application of the new EOS was demonstrated by calculating the solubility (sorption) of CO2 in polymers, the extent of swelling of polymers due to CO2, and the solubility of polymers in CO2 (cloud points). Both sorption and cloud point behavior in CO2 + polymer systems could be calculated using a single value of for each binary system. Ionic Liquids (ILs) can also incorporate electron donating functional groups in their structure. Evidence for the interaction of such ILs with CO2 can be found in the large values of the enthalpies of absorption of CO2 in these ILs. The ALF EOS was therefore extended to CO2 + IL systems using the enthalpy of absorption as a measure of association ( ) in these systems. was again treated as an adjustable parameter in the calculation of the CO2 solubility in ILs. A single value of was sufficient to predict swelling in these systems within experimental error.

Topological order is a new field of research involving exotic physics. Among other things it has been suggested as a means for realising fault-tolerant quantum computation. Topological degeneracy, i.e. the ground state degeneracy of a topologically ordered state, is one of the quantities that have been used to characterize such states. Topological order has also been suggested as a possible quantum information storage. We study two-dimensional lattice models defined on a closed manifold, specifically on a torus, and find that these systems exhibit topological degeneracy proportional to the genus of the manifold on which they are defined. We also find that the addition of lattice dislocations increases the ground state degeneracy, a behaviour that can be interpreted as artificially increasing the genus of the manifold. We derive the fusion and braiding rules of the model, which are then used to calculate the braiding properties of the dislocations themselves. These turn out to resemble non-abelian anyons, a property that is important for the possibility to achieve universal quantum computation. One can also emulate lattice dislocations synthetically, by adding an external field. This makes them more realistic for potential experimental realisations. / Topologisk ordning är ett nytt område inom fysik som bland annat verkar lovande som verktyg för förverkligandet av kvantdatorer. En av storheterna som karakteriserar topologiska tillstånd är det totala antalet degenererade grundtillstånd, den topologiska degenerationen. Topologisk ordning har också föreslagits som ett möjligt sätt att lagra kvantdata. Vi undersöker tvådimensionella gittermodeller definierade på en sluten mångfald, specifikt en torus, och finner att dessa system påvisar topologisk degeneration som är proportionerlig mot mångfaldens topologiska genus. När dislokationer introduceras i gittret finner vi att grundtillståndets degeneration ökar, något som kan ses som en artificiell ökning av mångfaldens genus. Vi härleder sammanslagningsregler och flätningsregler för modellen och använder sedan dessa för att räkna ut flätegenskaperna hos själva dislokationerna. Dessa visar sig likna icke-abelska anyoner, en egenskap som är viktiga för möjligheten att kunna utföra universella kvantberäkningar. Det går också att emulera dislokationer i gittret genom att lägga på ett yttre fält. Detta gör dem mer realistiska för eventuella experimentella realisationer.

Hydraulic fracturing is a multi-physics multi-scale problem related to natural processes such as the formation of dikes. It also has wide engineering applications such as extraction of unconventional resources, enhanced geothermal energy and carbon capture and storage. Current simulators are highly simplified because of the assumption of homogeneous reservoir. Unconventional reservoirs are heterogeneous owing to the presence of natural fracture network. Because of high computational effort, three-dimensional multi-scale simulations are uncommon, in particular, modelling material as a heterogeneous medium. Lattice Element Method (LEM) is therefore proposed for multi-scale simulation of heterogeneous material. In LEM, material is discretised into cells and their interactions are modelled by lattices, hence a three-dimensional model is simplified to a network of one-dimensional lattice. Normal, shear and rotational springs are used to define the constitutive laws of a lattice. LEM enables desktop computers for simulation of a lattice model that consists of millions of lattices. From simulations, normal springs govern the macroscopic bulk deformation while shear springs govern the macroscopic distortion. There is fluctuation of stresses even under uniform loading which is one of the characteristics of a lattice model. The magnitude increases with the stiffness ratio of shear spring to normal spring. Fracturing process can be modelled by LEM by introducing a microscopic tensile strength and a microscopic shear strength to the lattice properties. The strength parameters can be related to fracture toughness with the length scales of cells. From simulations, the relationships between model parameters and macroscopic parameters that are measurable in experiments are identified. From the simulations of uni-axial tension tests, both the spring stiffness ratio and the applied heterogeneity govern the fracturing process. The heterogeneity increases the ductility at the expense of the reduction on the macroscopic strengths. Different stages of fracturing are identified which are characterised by the model heterogeneity. Heterogeneous models go through the stages of the spatially distributed microscrack formation, the growth of multiple fracture clusters to the dominant fracture propagation. For homogeneous models, one of the microcracks rapidly propagates and becomes a dominant fracture with the absence of intermediate stages. From the uni-axial compression test simulations, the peak compressive stress is reached at the onset of the microscopic shear crack formation. Ductility is governed by the stiffness reduction ratio of a lattice in closed fractured stage to its unfractured stage. A novel Dual Lattice Model (DLM) is proposed for hydraulic fracture simulation by coupling a solid lattice model with a fluid lattice model. From DLM simulations of hydraulic fracturing of the classical penny shape crack problem under hydrostatic condition, the heterogeneities from both the fracture asperity and the applied heterogeneity increase the apparent fracture toughness. A semi-analytical solution is derived to consider the effect of fluid viscosity in the elastic deformation regime. Two asymptotes are identified that gives steep pressure gradients near the injection point and near the fracture tip which are also identified in the DLM simulations. Simulations also show three evolving regimes on energy dissipation/transfer mechanisms: the viscosity dominant, the elastic deformation dominant and the mixture of elastic deformation and toughness.