Synthesis, properties and application of polyphenylene phosphonium salts

The central objective of this work was to generate weakly coordinating cations of unprecedented molecular size providing an inherently stable hydrophobic shell around a central charge. It was hypothesized that divergent dendritic growth by means of thermal [4+2] Diels-Alder cycloaddition might represent a feasible synthetic method to circumvent steric constraints and enable a drastic increase in cation size.rnThis initial proposition could be verified: applying the divergent dendrimer synthesis to an ethynyl-functionalized tetraphenylphosphonium derivative afforded monodisperse cations with precisely nanoscopic dimensions for the first time. Furthermore, the versatile nature of the applied cascade reactions enabled a throughout flexible design and structural tuning of the desired target cations. The specific surface functionalization as well as the implementation of triazolyl-moieties within the dendrimer scaffold could be addressed by sophisticated variation of the employed building block units (see chapter 3). rnDue to the steric screening provided by their large, hydrophobic and shape-persistent polyphenylene shells, rigidly dendronized cations proved more weakly coordinating compared to their non-dendronized analogues. This hypothesis has been experimentally confirmed by means of dielectric spectroscopy (see chapter 4). It was demonstrated for a series of dendronized borate salts that the degree of ion dissociation increased with the size of the cations. The utilization of the very large phosphonium cations developed within this work almost achieved to separate the charge carriers about the Bjerrum length in solvents of low polarity, which was reflected by approaching near quantitative ion dissociation even at room temperature. In addition to effect the electrolyte behavior in solution, the steric enlargement of ions could be visualized by means of several crystal structure analyses. Thus an insight into lattice packing under the effect of extraordinary large cations could be gathered. rnAn essential theme of this work focused on the application of benzylphosphonium salts in the classical Wittig reaction, where the concept of dendronization served as synthetic means to introduce an exceptionally large polyphenylene substituent at the -position. The straightforward influence of this unprecedented bulky group on the Wittig stereochemistry was investigated by NMR-analysis of the resulting alkenes. Based on the obtained data a valuable explanation for the origin of the observed selectivity was brought in line with the up-to-date operating [2+2] cycloaddition mechanism. Furthermore, a reliable synthesis protocol for unsymmetrically substituted polyphenylene alkenes and stilbenes was established by the design of custom-built polyphenylene precursors (see chapter 5).rnFinally, fundamental experiments to functionalize a polymer chain with sterically shielded ionic groups either in the pending or internal position were outlined within this work. Thus, inherently hydrophobic polysalts shall be formed so that future research can invesigate their physical properties with regard to counter ion condensation and charge carrier mobility.rnIn summary, this work demonstrates how the principles of dendrimer chemistry can be applied to modify and specifically tailor the properties of salts. The numerously synthesized dendrimer-ions shown herein represent a versatile interface between classic organic and inorganic electrolytes, and defined macromolecular structures in the nanometer-scale. Furthermore the particular value of polyphenylene dendrimers in terms of a broad applicability was illustrated. This work accomplished in an interdisciplinary manner to give answer to various questions such as structural modification of ions, the resulting influence on the electrolyte behavior, as well as the stereochemical control of organic syntheses via polyphenylene phosphonium salts. rn

Wasserstoffbrücken-gebundene Netzwerke

Die supramolekularen Organisationen von 5-n-Alkoxyisophthaläuren und 2,5-Di-n-alkoxyterephthalsäuren bilden durch sekundäre Wechselwirkungen in Form van der Waals-Kräften und Wasserstoffbrückenbindungen hochgeordnete lamellare Strukturen, die sowohl mittels Röntgen-kristallgraphie als auch durch Rastertunnelmikroskopie (STM) visualisierbar sind. Diese Art der Aggregationsmuster ermöglicht den potentiell reaktiven Gruppen (Diacetylene, Zimtsäuren) in der Alkoxykette, einen optimalen Abstand und geometrische Orientierungen im Kristall einzunehmen, um lichtinduzierte Reaktionen im Festkörper durchzuführen. Einführung einer Amidgruppe an die Alkoxykette erweitert dieses Konzept, durch Erhalt einer zusätzlichen Wasserstoffbrückenbindung im hydrophoben Bereich der Organisation.Die Stabilität solche supramolekularer Aggregate kann durch Einführung semifluorierter Alkoxyketten stark beeinflußt werden, da die Inkompatibilität der pefluorierten Gruppen durch viele organische Verbindungen das Gleichgewicht nicht-kovalenter Wechselwirkung drastisch verändert. Diese Eigenschaft der semifluorierten 5-n-Alkoxyisopthalsäuren und 2,5-Di-n-alkoxyterephthalsäuren zeigt sich erstaunlicherweise jedoch nur bei einem großen Anteil des perfluorierten Alkylteils als der des nicht-perfluorierten. Da sich dann die perfluorierten Alkylteile untereinander organisieren können, erscheint die Änderung der Assoziationsstruktur von interdigitiert-lamellar zu nicht-interdigitiert-lamellar plausibel.Das gezielte Design eines neuen Organisationsmusters der 5-n-Alkoxyisophthalsäuren gelang durch Brechung der Symmetrie dieses Bausteins. Die Einführung zweier Alkoxyketten ließ die zweidimensionale Anordnung der wasserstoffbrückengebundenen Aromate unverändert, resultierte aber mit Ausbruch einer Alkoxykette aus dieser Ebene.

Strukturfaktoren, Modenkopplungsgleichungen und Glasübergang molekularer Kristalle

In dieser Arbeit wird der Orientierungsglasübergang ungeordneter, molekularer Kristalle untersucht. Die theoretische Behandlung ist durch die Anisotropie der Einteilchen-Verteilungsfunktion und der Paarfunktionen erschwert. Nimmt man ein starres Gitter, wird der reziproke Raum im Gegenzug auf die 1. Brillouin-Zone eingeschränkt. Der Orientierungsglasübergang wird im Rahmen der Modenkopplungsgleichungen studiert, die dazu hergeleitet werden. Als Modell dienen harte Rotationsellipsoide auf einem starren sc Gitter. Zur Berechnung der statischen tensoriellen Strukturfaktoren wird die Ornstein-Zernike(OZ)-Gleichung molekularer Kristalle abgeleitet und selbstkonsistent zusammen mit der von molekularen Flüssigkeiten übernommenen Percus-Yevick(PY)-Näherung gelöst. Parallel dazu werden die Strukturfaktoren durch MC-Simulationen ermittelt. Die OZ-Gleichung molekularer Kristalle ähnelt der von Flüssigkeiten, direkte und totale Korrelationsfunktion kommen jedoch wegen des starren Gitters nur ohne Konstantanteile in den Winkelvariablen vor, im Gegensatz zur PY-Näherung. Die Anisotropie bringt außerdem einen nichttrivialen Zusatzfaktor. OZ/PY-Strukturfaktoren und MC-Ergebnisse stimmen gut überein. Bei den Matrixelementen der Dichte-Dichte-Korrelationsfunktion gibt es drei Hauptverläufe: oszillatorisch, monoton und unregelmäßig abfallend. Oszillationen gehören zu alternierenden Dichtefluktuationen, führen zu Maxima der Strukturfaktoren am Zonenrand und kommen bei oblaten und genügend breiten prolaten, schwächer auch bei dünnen, nicht zu langen prolaten Ellipsoiden vor. Der exponentielle monotone Abfall kommt bei allen Ellipsoiden vor und führt zu Maxima der Strukturfaktoren in der Zonenmitte, was die Tendenz zu nematischer Ordnung zeigt. Die OZ/PY-Theorie ist durch divergierende Maxima der Strukturfaktoren begrenzt. Bei den Modenkopplungsgleichungen molekularer Kristalle zeigt sich eine große Ähnlichkeit mit denen molekularer Flüssigkeiten, jedoch spielen auf einem starrem Gitter nur die Matrixelemente mit l,l' > 0 eine Rolle und es finden Umklapps von reziproken Vektoren statt. Die Anisotropie bringt auch hier nichtkonstante Zusatzfaktoren ins Spiel. Bis auf flache oblate Ellipsoide wird die Modenkopplungs-Glaslinie von der Divergenz der Strukturfaktoren bestimmt. Für sehr lange Ellipsoide müssen die Strukturfaktoren zur Divergenz hin extrapoliert werden. Daher treibt nicht der Orientierungskäfigeffekt den Glasübergang, sondern Fluktuationen an einer Phasengrenze. Nahe der Kugelform ist keine zuverlässige Glasline festlegbar. Die eingefrorenen kritischen Dichte-Dichte-Korrelatoren haben nur in wenigen Fällen die Oszillationen der statischen Korrelatoren. Der monotone Abfall bleibt dagegen für lange Zeiten meist erhalten. Folglich haben die kritischen Modenkopplungs-Nichtergodizitätsparameter abgeschwächte Maxima in der Zonenmitte, während die Maxima am Zonenrand meist verschwunden sind. Die normierten Nichtergodizitätsparameter zeigen eine Fülle von Verläufen, besonders tiefer im Glas.

Lattice preferred orientation in omphacite: Examples from the Tauern Window, Austria and the Western Gneiss Region, Norway

It is lively debated how eclogites find their way from deep to mid-crustal levels during exhumation. Different exhumation models for high-pressure and ultrahigh-pressure rocks were suggested in previous studies, based mainly on field observations and less on microstructural studies on the exhumed rocks. The development and improvement of electron microscopy techniques allows it, to focus interest on direct investigations of microstructures and crystallographic properties in eclogites. In this case, it is of importance to study the applicability of crystallographic measurements on eclogites for exhumation processes and to unravel which processes affect eclogite textures. Previous studies suggested a strong relationship between deformation and lattice preferred orientation (LPO) in omphacite but it is still unclear if the deformation is related to the exhumation of eclogites. This study is focused on the questions which processes affect omphacite LPO and if textural investigations of omphacite are applicable for studying eclogite exhumation. Therefore, eclogites from two examples in the Alps and in the Caledonides were collected systematically and investigated with respect to omphacite LPO by using the electron backscattered diffraction (EBSD) technique. Omphacite textures of the Tauern Window (Austria) and the Western Gneiss Region (Norway) were studied to compare lattice preferred orientation with field observations and suggested exhumation models from previous studies. The interpretation of omphacite textures, regarding the deformation regime is mainly based on numerical simulations in previous studies. Omphacite LPO patterns of the Eclogite Zone are clearly independent from any kind of exhumation process. The textures were generated during omphacite growth on the prograde path of eclogite development until metamorphic peak conditions. Field observations in the Eclogite Zone show that kinematics in garnet mica schist, surrounding the eclogites, strongly indicate an extrusion wedge geometry. Stretching lineations show top-N thrusting at the base and a top-S normal faulting with a sinistral shear component at the top of the Eclogite Zone. The different shear sense on both sides of the unit does not affect the omphacite textures in any way. The omphacite lattice preferred orientation patterns of the Western Gneiss Region can not be connected with any exhumation model. The textures were probably generated during the metamorphic peak and reflect the change from subduction to exhumation. Eclogite Zone and Western Gneiss Region differ significantly in size and especially in metamorphic conditions. While the Eclogite Zone is characterized by constant P-T conditions (600-650°C, 20-25 kbar), the Western Gneiss Region contains a wide P-T range from high- to ultrahigh pressure conditions (400-800°C, 20-35 kbar). In contrast to this, the omphacite textures of both units are very similar. This means that omphacite LPO is independent from P-T conditions and therefore from burial depth. Further, in both units, omphacite LPO is independent from grain and subgrain size as well as from any shape preferred orientation (SPO) on grain and subgrain scale. Overall, omphacite lattice preferred orientation are generated on the prograde part of omphacite development. Therefore, textural investigations on omphacite LPO are not applicable to study eclogite exhumation.

Thermal fluctuations and boundary conditions in the lattice Boltzmann method

The lattice Boltzmann method is a popular approach for simulating hydrodynamic interactions in soft matter and complex fluids. The solvent is represented on a discrete lattice whose nodes are populated by particle distributions that propagate on the discrete links between the nodes and undergo local collisions. On large length and time scales, the microdynamics leads to a hydrodynamic flow field that satisfies the Navier-Stokes equation. In this thesis, several extensions to the lattice Boltzmann method are developed. In complex fluids, for example suspensions, Brownian motion of the solutes is of paramount importance. However, it can not be simulated with the original lattice Boltzmann method because the dynamics is completely deterministic. It is possible, though, to introduce thermal fluctuations in order to reproduce the equations of fluctuating hydrodynamics. In this work, a generalized lattice gas model is used to systematically derive the fluctuating lattice Boltzmann equation from statistical mechanics principles. The stochastic part of the dynamics is interpreted as a Monte Carlo process, which is then required to satisfy the condition of detailed balance. This leads to an expression for the thermal fluctuations which implies that it is essential to thermalize all degrees of freedom of the system, including the kinetic modes. The new formalism guarantees that the fluctuating lattice Boltzmann equation is simultaneously consistent with both fluctuating hydrodynamics and statistical mechanics. This establishes a foundation for future extensions, such as the treatment of multi-phase and thermal flows. An important range of applications for the lattice Boltzmann method is formed by microfluidics. Fostered by the "lab-on-a-chip" paradigm, there is an increasing need for computer simulations which are able to complement the achievements of theory and experiment. Microfluidic systems are characterized by a large surface-to-volume ratio and, therefore, boundary conditions are of special relevance. On the microscale, the standard no-slip boundary condition used in hydrodynamics has to be replaced by a slip boundary condition. In this work, a boundary condition for lattice Boltzmann is constructed that allows the slip length to be tuned by a single model parameter. Furthermore, a conceptually new approach for constructing boundary conditions is explored, where the reduced symmetry at the boundary is explicitly incorporated into the lattice model. The lattice Boltzmann method is systematically extended to the reduced symmetry model. In the case of a Poiseuille flow in a plane channel, it is shown that a special choice of the collision operator is required to reproduce the correct flow profile. This systematic approach sheds light on the consequences of the reduced symmetry at the boundary and leads to a deeper understanding of boundary conditions in the lattice Boltzmann method. This can help to develop improved boundary conditions that lead to more accurate simulation results.

Interacting bosons and fermions in three-dimensional optical lattice potentials

This thesis reports on the realization, characterization and analysis of ultracold bosonic and fermionic atoms in three-dimensional optical lattice potentials. Ultracold quantum gases in optical lattices can be regarded as ideal model systems to investigate quantum many-body physics. In this work interacting ensembles of bosonic 87Rb and fermionic 40K atoms are employed to study equilibrium phases and nonequilibrium dynamics. The investigations are enabled by a versatile experimental setup, whose core feature is a blue-detuned optical lattice that is combined with Feshbach resonances and a red-detuned dipole trap to allow for independent control of tunneling, interactions and external confinement. The Fermi-Hubbard model, which plays a central role in the theoretical description of strongly correlated electrons, is experimentally realized by loading interacting fermionic spin mixtures into the optical lattice. Using phase-contrast imaging the in-situ size of the atomic density distribution is measured, which allows to extract the global compressibility of the many-body state as a function of interaction and external confinement. Thereby, metallic and insulating phases are clearly identified. At strongly repulsive interaction, a vanishing compressibility and suppression of doubly occupied lattice sites signal the emergence of a fermionic Mott insulator. In a second series of experiments interaction effects in bosonic lattice quantum gases are analyzed. Typically, interactions between microscopic particles are described as two-body interactions. As such they are also contained in the single-band Bose-Hubbard model. However, our measurements demonstrate the presence of multi-body interactions that effectively emerge via virtual transitions of atoms to higher lattice bands. These findings are enabled by the development of a novel atom optical measurement technique: In quantum phase revival spectroscopy periodic collapse and revival dynamics of the bosonic matter wave field are induced. The frequencies of the dynamics are directly related to the on-site interaction energies of atomic Fock states and can be read out with high precision. The third part of this work deals with mixtures of bosons and fermions in optical lattices, in which the interspecies interactions are accurately controlled by means of a Feshbach resonance. Studies of the equilibrium phases show that the bosonic superfluid to Mott insulator transition is shifted towards lower lattice depths when bosons and fermions interact attractively. This observation is further analyzed by applying quantum phase revival spectroscopy to few-body systems consisting of a single fermion and a coherent bosonic field on individual lattice sites. In addition to the direct measurement of Bose-Fermi interaction energies, Bose-Bose interactions are proven to be modified by the presence of a fermion. This renormalization of bosonic interaction energies can explain the shift of the Mott insulator transition. The experiments of this thesis lay important foundations for future studies of quantum magnetism with fermionic spin mixtures as well as for the realization of complex quantum phases with Bose-Fermi mixtures. They furthermore point towards physics that reaches beyond the single-band Hubbard model.

Charge-transport simulations in organic semiconductors

In this thesis we have extended the methods for microscopic charge-transport simulations for organic semiconductors. In these materials the weak intermolecular interactions lead to spatially localized charge carriers, and the charge transport occurs as an activated hopping process between diabatic states. In addition to weak electronic couplings between these states, different electrostatic environments in the organic material lead to a broadening of the density of states for the charge energies which limits carrier mobilities.rnThe contributions to the method development includern(i) the derivation of a bimolecular charge-transfer rate,rn(ii) the efficient evaluation of intermolecular (outer-sphere) reorganization energies,rn(iii) the investigation of effects of conformational disorder on intramolecular reorganization energies or internal site energiesrnand (iv) the inclusion of self-consistent polarization interactions for calculation of charge energies.These methods were applied to study charge transport in amorphous phases of small molecules used in the emission layer of organic light emitting diodes (OLED).rnWhen bulky substituents are attached to an aromatic core in order to adjust energy levels or prevent crystallization, a small amount of delocalization of the frontier orbital to the substituents can increase electronic couplings between neighboring molecules. This leads to improved charge-transfer rates and, hence, larger charge-mobility. We therefore suggest using the mesomeric effect (as opposed to the inductive effect) when attaching substituents to aromatic cores, which is necessary for example in deep blue OLEDs, where the energy levels of a host molecule have to be adjusted to those of the emitter.rnFurthermore, the energy landscape for charges in an amorphous phase cannot be predicted by mesoscopic models because they approximate the realistic morphology by a lattice and represent molecular charge distributions in a multipole expansion. The microscopic approach shows that a polarization-induced stabilization of a molecule in its charged and neutral states can lead to large shifts, broadening, and traps in the distribution of charge energies. These results are especially important for multi-component systems (the emission layer of an OLED or the donor-acceptor interface of an organic solar cell), if the change in polarizability upon charging (or excitation in case of energy transport) is different for the components. Thus, the polarizability change upon charging or excitation should be added to the set of molecular parameters essential for understanding charge and energy transport in organic semiconductors.rnWe also studied charge transport in self-assembled systems, where intermolecular packing motives induced by side chains can increase electronic couplings between molecules. This leads to larger charge mobility, which is essential to improve devices such as organic field effect transistors, where low carrier mobilities limit the switching frequency.rnHowever, it is not sufficient to match the average local molecular order induced by the sidernchains (such as the pitch angle between consecutive molecules in a discotic mesophase) with maxima of the electronic couplings.rnIt is also important to make the corresponding distributions as narrow as possible compared to the window determined by the closest minima of thernelectronic couplings. This is especially important in one-dimensional systems, where charge transport is limited by the smallest electronic couplings.rnThe immediate implication for compound design is that the side chains should assist the self-assemblingrnprocess not only via soft entropic interactions, but also via stronger specific interactions, such as hydrogen bonding.rnrnrnrn

Packing a trunk : a physically based approach with full motional freedom

Geometric packing problems may be formulated mathematically as constrained optimization problems. But finding a good solution is a challenging task. The more complicated the geometry of the container or the objects to be packed, the more complex the non-penetration constraints become. In this work we propose the use of a physics engine that simulates a system of colliding rigid bodies. It is a tool to resolve interpenetration conflicts and to optimize configurations locally. We develop an efficient and easy-to-implement physics engine that is specialized for collision detection and contact handling. In succession of the development of this engine a number of novel algorithms for distance calculation and intersection volume were designed and imple- mented, which are presented in this work. They are highly specialized to pro- vide fast responses for cuboids and triangles as input geometry whereas the concepts they are based on can easily be extended to other convex shapes. Especially noteworthy in this context is our ε-distance algorithm - a novel application that is not only very robust and fast but also compact in its im- plementation. Several state-of-the-art third party implementations are being presented and we show that our implementations beat them in runtime and robustness. The packing algorithm that lies on top of the physics engine is a Monte Carlo based approach implemented for packing cuboids into a container described by a triangle soup. We give an implementation for the SAE J1100 variant of the trunk packing problem. We compare this implementation to several established approaches and we show that it gives better results in faster time than these existing implementations.

Random lattice models

This thesis deals with three different physical models, where each model involves a random component which is linked to a cubic lattice. First, a model is studied, which is used in numerical calculations of Quantum Chromodynamics.In these calculations random gauge-fields are distributed on the bonds of the lattice. The formulation of the model is fitted into the mathematical framework of ergodic operator families. We prove, that for small coupling constants, the ergodicity of the underlying probability measure is indeed ensured and that the integrated density of states of the Wilson-Dirac operator exists. The physical situations treated in the next two chapters are more similar to one another. In both cases the principle idea is to study a fermion system in a cubic crystal with impurities, that are modeled by a random potential located at the lattice sites. In the second model we apply the Hartree-Fock approximation to such a system. For the case of reduced Hartree-Fock theory at positive temperatures and a fixed chemical potential we consider the limit of an infinite system. In that case we show the existence and uniqueness of minimizers of the Hartree-Fock functional. In the third model we formulate the fermion system algebraically via C*-algebras. The question imposed here is to calculate the heat production of the system under the influence of an outer electromagnetic field. We show that the heat production corresponds exactly to what is empirically predicted by Joule's law in the regime of linear response.

Hadronic matrix elements in lattice QCD

The lattice formulation of Quantum ChromoDynamics (QCD) has become a reliable tool providing an ab initio calculation of low-energy quantities. Despite numerous successes, systematic uncertainties, such as discretisation effects, finite-size effects, and contaminations from excited states, are inherent in any lattice calculation. Simulations with controlled systematic uncertainties and close to the physical pion mass have become state-of-the-art. We present such a calculation for various hadronic matrix elements using non-perturbatively O(a)-improved Wilson fermions with two dynamical light quark flavours. The main topics covered in this thesis are the axial charge of the nucleon, the electro-magnetic form factors of the nucleon, and the leading hadronic contributions to the anomalous magnetic moment of the muon. Lattice simulations typically tend to underestimate the axial charge of the nucleon by 5 − 10%. We show that including excited state contaminations using the summed operator insertion method leads to agreement with the experimentally determined value. Further studies of systematic uncertainties reveal only small discretisation effects. For the electro-magnetic form factors of the nucleon, we see a similar contamination from excited states as for the axial charge. The electro-magnetic radii, extracted from a dipole fit to the momentum dependence of the form factors, show no indication of finite-size or cutoff effects. If we include excited states using the summed operator insertion method, we achieve better agreement with the radii from phenomenology. The anomalous magnetic moment of the muon can be measured and predicted to very high precision. The theoretical prediction of the anomalous magnetic moment receives contribution from strong, weak, and electro-magnetic interactions, where the hadronic contributions dominate the uncertainties. A persistent 3σ tension between the experimental determination and the theoretical calculation is found, which is considered to be an indication for physics beyond the Standard Model. We present a calculation of the connected part of the hadronic vacuum polarisation using lattice QCD. Partially twisted boundary conditions lead to a significant improvement of the vacuum polarisation in the region of small momentum transfer, which is crucial in the extraction of the hadronic vacuum polarisation.

Imaging and modeling of pore scale processes in porous media using X-ray computed tomography and lattice Boltzmann solver

In der Erdöl– und Gasindustrie sind bildgebende Verfahren und Simulationen auf der Porenskala im Begriff Routineanwendungen zu werden. Ihr weiteres Potential lässt sich im Umweltbereich anwenden, wie z.B. für den Transport und Verbleib von Schadstoffen im Untergrund, die Speicherung von Kohlendioxid und dem natürlichen Abbau von Schadstoffen in Böden. Mit der Röntgen-Computertomografie (XCT) steht ein zerstörungsfreies 3D bildgebendes Verfahren zur Verfügung, das auch häufig für die Untersuchung der internen Struktur geologischer Proben herangezogen wird. Das erste Ziel dieser Dissertation war die Implementierung einer Bildverarbeitungstechnik, die die Strahlenaufhärtung der Röntgen-Computertomografie beseitigt und den Segmentierungsprozess dessen Daten vereinfacht. Das zweite Ziel dieser Arbeit untersuchte die kombinierten Effekte von Porenraumcharakteristika, Porentortuosität, sowie die Strömungssimulation und Transportmodellierung in Porenräumen mit der Gitter-Boltzmann-Methode. In einer zylindrischen geologischen Probe war die Position jeder Phase auf Grundlage der Beobachtung durch das Vorhandensein der Strahlenaufhärtung in den rekonstruierten Bildern, das eine radiale Funktion vom Probenrand zum Zentrum darstellt, extrahierbar und die unterschiedlichen Phasen ließen sich automatisch segmentieren. Weiterhin wurden Strahlungsaufhärtungeffekte von beliebig geformten Objekten durch einen Oberflächenanpassungsalgorithmus korrigiert. Die Methode der „least square support vector machine” (LSSVM) ist durch einen modularen Aufbau charakterisiert und ist sehr gut für die Erkennung und Klassifizierung von Mustern geeignet. Aus diesem Grund wurde die Methode der LSSVM als pixelbasierte Klassifikationsmethode implementiert. Dieser Algorithmus ist in der Lage komplexe geologische Proben korrekt zu klassifizieren, benötigt für den Fall aber längere Rechenzeiten, so dass mehrdimensionale Trainingsdatensätze verwendet werden müssen. Die Dynamik von den unmischbaren Phasen Luft und Wasser wird durch eine Kombination von Porenmorphologie und Gitter Boltzmann Methode für Drainage und Imbibition Prozessen in 3D Datensätzen von Böden, die durch synchrotron-basierte XCT gewonnen wurden, untersucht. Obwohl die Porenmorphologie eine einfache Methode ist Kugeln in den verfügbaren Porenraum einzupassen, kann sie dennoch die komplexe kapillare Hysterese als eine Funktion der Wassersättigung erklären. Eine Hysterese ist für den Kapillardruck und die hydraulische Leitfähigkeit beobachtet worden, welche durch die hauptsächlich verbundenen Porennetzwerke und der verfügbaren Porenraumgrößenverteilung verursacht sind. Die hydraulische Konduktivität ist eine Funktion des Wassersättigungslevels und wird mit einer makroskopischen Berechnung empirischer Modelle verglichen. Die Daten stimmen vor allem für hohe Wassersättigungen gut überein. Um die Gegenwart von Krankheitserregern im Grundwasser und Abwässern vorhersagen zu können, wurde in einem Bodenaggregat der Einfluss von Korngröße, Porengeometrie und Fluidflussgeschwindigkeit z.B. mit dem Mikroorganismus Escherichia coli studiert. Die asymmetrischen und langschweifigen Durchbruchskurven, besonders bei höheren Wassersättigungen, wurden durch dispersiven Transport aufgrund des verbundenen Porennetzwerks und durch die Heterogenität des Strömungsfeldes verursacht. Es wurde beobachtet, dass die biokolloidale Verweilzeit eine Funktion des Druckgradienten als auch der Kolloidgröße ist. Unsere Modellierungsergebnisse stimmen sehr gut mit den bereits veröffentlichten Daten überein.

Hadronic correlation functions with quark-disconnected contributions in lattice QCD

One of the fundamental interactions in the Standard Model of particle physicsrnis the strong force, which can be formulated as a non-abelian gauge theoryrncalled Quantum Chromodynamics (QCD). rnIn the low-energy regime, where the QCD coupling becomes strong and quarksrnand gluons are confined to hadrons, a perturbativernexpansion in the coupling constant is not possible.rnHowever, the introduction of a four-dimensional Euclidean space-timernlattice allows for an textit{ab initio} treatment of QCD and provides arnpowerful tool to study the low-energy dynamics of hadrons.rnSome hadronic matrix elements of interest receive contributionsrnfrom diagrams including quark-disconnected loops, i.e. disconnected quarkrnlines from one lattice point back to the same point. The calculation of suchrnquark loops is computationally very demanding, because it requires knowledge ofrnthe all-to-all propagator. In this thesis we use stochastic sources and arnhopping parameter expansion to estimate such propagators.rnWe apply this technique to study two problems which relay crucially on therncalculation of quark-disconnected diagrams, namely the scalar form factor ofrnthe pion and the hadronic vacuum polarization contribution to the anomalousrnmagnet moment of the muon.rnThe scalar form factor of the pion describes the coupling of a charged pion torna scalar particle. We calculate the connected and the disconnected contributionrnto the scalar form factor for three different momentum transfers. The scalarrnradius of the pion is extracted from the momentum dependence of the form factor.rnThe use ofrnseveral different pion masses and lattice spacings allows for an extrapolationrnto the physical point. The chiral extrapolation is done using chiralrnperturbation theory ($chi$PT). We find that our pion mass dependence of thernscalar radius is consistent with $chi$PT at next-to-leading order.rnAdditionally, we are able to extract the low energy constant $ell_4$ from thernextrapolation, and ourrnresult is in agreement with results from other lattice determinations.rnFurthermore, our result for the scalar pion radius at the physical point isrnconsistent with a value that was extracted from $pipi$-scattering data. rnThe hadronic vacuum polarization (HVP) is the leading-order hadronicrncontribution to the anomalous magnetic moment $a_mu$ of the muon. The HVP canrnbe estimated from the correlation of two vector currents in the time-momentumrnrepresentation. We explicitly calculate the corresponding disconnectedrncontribution to the vector correlator. We find that the disconnectedrncontribution is consistent with zero within its statistical errors. This resultrncan be converted into an upper limit for the maximum contribution of therndisconnected diagram to $a_mu$ by using the expected time-dependence of therncorrelator and comparing it to the corresponding connected contribution. Wernfind the disconnected contribution to be smaller than $approx5%$ of thernconnected one. This value can be used as an estimate for a systematic errorrnthat arises from neglecting the disconnected contribution.rn