Numerical and analytical approaches to strongly correlated electron systems

In this thesis we consider three different models for strongly correlated electrons, namely a multi-band Hubbard model as well as the spinless Falicov-Kimball model, both with a semi-elliptical density of states in the limit of infinite dimensions d, and the attractive Hubbard model on a square lattice in d=2. In the first part, we study a two-band Hubbard model with unequal bandwidths and anisotropic Hund's rule coupling (J_z-model) in the limit of infinite dimensions within the dynamical mean-field theory (DMFT). Here, the DMFT impurity problem is solved with the use of quantum Monte Carlo (QMC) simulations. Our main result is that the J_z-model describes the occurrence of an orbital-selective Mott transition (OSMT), in contrast to earlier findings. We investigate the model with a high-precision DMFT algorithm, which was developed as part of this thesis and which supplements QMC with a high-frequency expansion of the self-energy. The main advantage of this scheme is the extraordinary accuracy of the numerical solutions, which can be obtained already with moderate computational effort, so that studies of multi-orbital systems within the DMFT+QMC are strongly improved. We also found that a suitably defined Falicov-Kimball (FK) model exhibits an OSMT, revealing the close connection of the Falicov-Kimball physics to the J_z-model in the OSM phase. In the second part of this thesis we study the attractive Hubbard model in two spatial dimensions within second-order self-consistent perturbation theory. This model is considered on a square lattice at finite doping and at low temperatures. Our main result is that the predictions of first-order perturbation theory (Hartree-Fock approximation) are renormalized by a factor of the order of unity even at arbitrarily weak interaction (U->0). The renormalization factor q can be evaluated as a function of the filling n for 00, the q-factor vanishes, signaling the divergence of self-consistent perturbation theory in this limit. Thus we present the first asymptotically exact results at weak-coupling for the negative-U Hubbard model in d=2 at finite doping.

Numerical studies of colloidal systems

The interplay of hydrodynamic and electrostatic forces is of great importance for the understanding of colloidal dispersions. Theoretical descriptions are often based on the so called standard electrokinetic model. This Mean Field approach combines the Stokes equation for the hydrodynamic flow field, the Poisson equation for electrostatics and a continuity equation describing the evolution of the ion concentration fields. In the first part of this thesis a new lattice method is presented in order to efficiently solve the set of non-linear equations for a charge-stabilized colloidal dispersion in the presence of an external electric field. Within this framework, the research is mainly focused on the calculation of the electrophoretic mobility. Since this transport coefficient is independent of the electric field only for small driving, the algorithm is based upon a linearization of the governing equations. The zeroth order is the well known Poisson-Boltzmann theory and the first order is a coupled set of linear equations. Furthermore, this set of equations is divided into several subproblems. A specialized solver for each subproblem is developed, and various tests and applications are discussed for every particular method. Finally, all solvers are combined in an iterative procedure and applied to several interesting questions, for example, the effect of the screening mechanism on the electrophoretic mobility or the charge dependence of the field-induced dipole moment and ion clouds surrounding a weakly charged sphere. In the second part a quantitative data analysis method is developed for a new experimental approach, known as "Total Internal Reflection Fluorescence Cross-Correlation Spectroscopy" (TIR-FCCS). The TIR-FCCS setup is an optical method using fluorescent colloidal particles to analyze the flow field close to a solid-fluid interface. The interpretation of the experimental results requires a theoretical model, which is usually the solution of a convection-diffusion equation. Since an analytic solution is not available due to the form of the flow field and the boundary conditions, an alternative numerical approach is presented. It is based on stochastic methods, i. e. a combination of a Brownian Dynamics algorithm and Monte Carlo techniques. Finally, experimental measurements for a hydrophilic surface are analyzed using this new numerical approach.

Higher-order perturbative relativistic corrections to energies and properties

Relativistic effects need to be considered in quantum-chemical calculations on systems including heavy elements or when aiming at high accuracy for molecules containing only lighter elements. In the latter case, consideration of relativistic effects via perturbation theory is an attractive option. Among the available techniques, Direct Perturbation Theory (DPT) in its lowest order (DPT2) has become a standard tool for the calculation of relativistic corrections to energies and properties.In this work, the DPT treatment is extended to the next order (DPT4). It is demonstrated that the DPT4 correction can be obtained as a second derivative of the energy with respect to the relativistic perturbation parameter. Accordingly, differentiation of a suitable Lagrangian, thereby taking into account all constraints on the wave function, provides analytic expressions for the fourth-order energy corrections. The latter have been implemented at the Hartree-Fock level and within second-order Møller-Plesset perturbaton theory using standard analytic second-derivative techniques into the CFOUR program package. For closed-shell systems, the DPT4 corrections consist of higher-order scalar-relativistic effects as well as spin-orbit corrections with the latter appearing here for the first time in the DPT series.Relativistic corrections are reported for energies as well as for first-order electrical properties and compared to results from rigorous four-component benchmark calculations in order to judge the accuracy and convergence of the DPT expansion for both the scalar-relativistic as well as the spin-orbit contributions. Additionally, the importance of relativistic effects to the bromine and iodine quadrupole-coupling tensors is investigated in a joint experimental and theoretical study concerning the rotational spectra of CH2BrF, CHBrF2, and CH2FI.

Darstellung von substituierten Phenylacetylenen und Untersuchung ihrer von Palladiumverbindungen und Phosphorliganden homogen katalysierten Dimerisierung

In dieser Arbeit werden Homodimerisierungen von substituierten Phenylacetylenen mit Butinsäureethylester untersucht. Als Katalysatorsysteme dienen Palladium-Phosphan-Systeme.Kinetische Untersuchungen dieser Reaktionen zeigen, dass im Allgemeinen keine einfache Reaktionsordnung vorliegt. Nur für die Dimerisierung von Phenylacetylen mit einem Katalysatorsystem aus Palladium(II)-acetat und Triphenylphosphan findet sich eine Reaktion erster Ordnung.Vergleiche von Produktzusammensetzungen nach Ablauf einiger Homodimerisierungen bei verschiedenen Temperaturen führen zu dem Schluss, dass diese Reaktionen durch Temperaturveränderungen nicht kinetisch kontrollierbar sind.Die Struktur des 2,4-Bis(4-methoxyphenyl)-but-1-en-3-ins, das bei der Dimerisierung von 4-Methoxy-phenylacetylen entsteht, konnte durch eine Röntgenstrukturanalyse bestimmt werden.

Synthese von flüssigkristallinen Polysiloxanen und Charakterisierung der smektischen Phasen

Gegenstand dieser Arbeit ist die Untersuchung der smektischen Phasen von Polysiloxanen mit flüssigkristallinen Seitengruppen (LC-Polysiloxane). Der erste Teil der vorliegenden Arbeit befasste sich mit der Herstellung verschiedener flüssigkristalliner ferroelektrischer Polysiloxane. Die Polymere wurden in Bezug auf das verwendete Polymerrückgrat (Homo- und Copolysiloxan) sowie durch den zusätzlichen Einbau von vernetzbaren Seitengruppen variiert. Im zweiten Teil der Arbeit wurden die Eigenschaften der smektischen Phasen der hergestellten Substanzen näher untersucht. Ein erster Untersuchungsgegenstand war das Dehnungsverhalten von freistehenden flüssigkristallinen Elastomerfilmen (LCE). Bei der Verwendung eines Polymers, in dem nur ein Teil des Polysiloxanrückgrats mit Seitengruppen substituiert ist, wird die uniaxiale Dehnung des Films parallel zu den smektischen Schichten durch eine gleichmäßige Kontraktion in der Filmebene und parallel zur Schichtnormalen ausgeglichen, was auf einen außergewöhnlich niedrigen smektischen Schichtkompressionsmodul zurückzuführen ist. Im Gegensatz dazu ist dieser Modul bei den Homopolymersystemen so groß, dass praktisch senkrecht zu den smektischen Schichten keine Kontraktion stattfindet. Ein zweiter Untersuchungsgegenstand der Netzwerkbildung bestand in der Bestimmung der dynamisch-mechanischen Eigenschaften der LC-Polysiloxane mittels eines Oszillationsrheometers. Hier erfolgten die Messungen von Speicher- und Verlustmodul in Abhängigkeit vom Polymerrückgrat und von der Vernetzung. Die unvernetzten Systeme zeigten in den smektischen Phasen (oberhalb Tg) noch – im wesentlichen – Festkörpereigenschaften (physikalische Vernetzung) mit einem dominierenden Speichermodul beim LC-Homopolysiloxan. Beim LC-Copolysiloxan haben beide Module eine gleiche Größenordnung. Am Phasenübergang in die isotrope Phase wurden beide Module absolut kleiner, der Verlustmodul aber relativ größer. In der isotropen Phase verhalten sich die LC-Polymere damit überwiegend wie viskose Schmelzen. Außerdem korrelierten die mittels DSC bestimmten Phasenübergangstemperaturen mit einer Änderung der dynamisch-mechanischen Eigenschaften. Nach der Vernetzung dominierte der Speichermodul sowohl beim LC-Homo- als auch beim LC-Copolysiloxan bis in die isotrope Phase, und es war aufgrund der Bildung einer festen Netzwerkstruktur keine Abhängigkeit der Module von Phasenübergängen mehr erkennbar. Als dritter Untersuchungsgegenstand wurde der Phasenübergang zwischen den beiden smektischen Phasen (SmC* nach SmA*) der flüssigkristallinen Polysiloxane näher behandelt. Als wichtigstes Ergebnis ist festzuhalten, dass die verdünnten LC-Polysiloxane an diesem Übergang fast keine Schichtdickenänderung aufweisen. Dazu wurde jeweils die röntgenographisch ermittelte Schichtdicke mit der aus den optischen Tiltwinkeln theoretisch berechneten Schichtdicke verglichen. Dadurch konnte gezeigt werden, dass sich die Phasenübergänge nach dem de Vries-Modell verhalten. Damit konnte zum ersten Mal an Polymersystemen ein de Vries-Verhalten nachgewiesen werden. Im Gegensatz dazu zeigte das Homopolysiloxan mit dem Dreiringmesogen beim Übergang von SmC* nach SmA* einen ausgeprägten Sprung in der Schichtdicke. Wie auch durch DSC-Messungen nachweisbar, lag ein Phasenübergang 1. Ordnung vor. Bei den LC-Copolysiloxanen liegt dagegen ein Phasenübergang 2. Ordnung vor. Schließlich wurde die Schichtdicke unter dem Einfluss der Vernetzung untersucht. Beim LC-Copolysiloxan mit dem Dreiringmesogen und einem Anteil an vernetzbaren Gruppen von 15 % wurde eine Stabilisierung der smektischen Phasen erreicht. Zum einen war die Änderung der Schichtdicke am SmC*-SmA*-Phasenübergang geringer im Vergleich zum unvernetzten System und zum anderen war noch 50 °C über der ursprünglichen Klärtemperatur eine smektische Schichtstruktur röntgenographisch nachzuweisen. Insgesamt ist es mit den verschiedenen Untersuchungsmethoden gelungen, einen systematischen Unterschied zwischen smektischen Homo- und Copolysiloxanen aufzuzeigen, der seinen Ursprung – aller Wahrscheinlichkeit nach – in der Mikrophasenseparation von Mesogenen und Polysiloxanketten findet.

Crystal phases and glassy dynamics in monodisperse hard ellipsoids

We have performed Monte Carlo and molecular dynamics simulations of suspensions of monodisperse, hard ellipsoids of revolution. Hard-particle models play a key role in statistical mechanics. They are conceptually and computationally simple, and they offer insight into systems in which particle shape is important, including atomic, molecular, colloidal, and granular systems. In the high density phase diagram of prolate hard ellipsoids we have found a new crystal, which is more stable than the stretched FCC structure proposed previously . The new phase, SM2, has a simple monoclinic unit cell containing a basis of two ellipsoids with unequal orientations. The angle of inclination is very soft for length-to-width (aspect) ratio l/w=3, while the other angles are not. A symmetric state of the unit cell exists, related to the densest-known packings of ellipsoids; it is not always the stable one. Our results remove the stretched FCC structure for aspect ratio l/w=3 from the phase diagram of hard, uni-axial ellipsoids. We provide evidence that this holds between aspect ratios 3 and 6, and possibly beyond. Finally, ellipsoids in SM2 at l/w=1.55 exhibit end-over-end flipping, warranting studies of the cross-over to where this dynamics is not possible. Secondly, we studied the dynamics of nearly spherical ellipsoids. In equilibrium, they show a first-order transition from an isotropic phase to a rotator phase, where positions are crystalline but orientations are free. When over-compressing the isotropic phase into the rotator regime, we observed super-Arrhenius slowing down of diffusion and relaxation, and signatures of the cage effect. These features of glassy dynamics are sufficiently strong that asymptotic scaling laws of the Mode-Coupling Theory of the glass transition (MCT) could be tested, and were found to apply. We found strong coupling of positional and orientational degrees of freedom, leading to a common value for the MCT glass-transition volume fraction. Flipping modes were not slowed down significantly. We demonstrated that the results are independent of simulation method, as predicted by MCT. Further, we determined that even intra-cage motion is cooperative. We confirmed the presence of dynamical heterogeneities associated with the cage effect. The transit between cages was seen to occur on short time scales, compared to the time spent in cages; but the transit was shown not to involve displacements distinguishable in character from intra-cage motion. The presence of glassy dynamics was predicted by molecular MCT (MMCT). However, as MMCT disregards crystallization, a test by simulation was required. Glassy dynamics is unusual in monodisperse systems. Crystallization typically intervenes unless polydispersity, network-forming bonds or other asymmetries are introduced. We argue that particle anisometry acts as a sufficient source of disorder to prevent crystallization. This sheds new light on the question of which ingredients are required for glass formation.

Mineral chemistry and structural relationships of inclusions in diamond samples

Diamant ist das härteste Mineral – und dazu ein Edelstein -, das unter höchstem Druck und hohen Temperaturen in tiefen kontinentalen Regionen der Erde kristallisiert. Die Mineraleinschlüsse in Diamanten werden durch die physikalische Stabilität und chemische Beständigkeit der umgebenden – eigentlich metastabilen -Diamant-Phase geschützt. Aufgrund der koexistierenden Phasenkombination ermöglichen sie, die Mineral-Entwicklung zu studieren, während deren der Einschlüssen und die Diamanten kristallisierten. rnDie Phasenkombinationen von Diamant und Chrom-Pyrop, Chrom-Diopsid, Chromit, Olivin, Graphit und Enstatit nebeneinander (teilweise in Berührungsexistenz) mit Chrom-Pyrop Einschlüssen wurden von neunundzwanzig Diamant-Proben von sechs Standorten in Südafrika (Premier, Koffiefontein, De Beers Pool, Finsch, Venetia und Koingnaas Minen) und Udachnaya (Sibirien/Russland) identifiziert und charakterisiert. Die Mineraleinschlüsse weisen z.T. kubo-oktaedrische Form auf, die unabhängig von ihren eigenen Kristallsystemen ausgebildet werden können. Das bedeutet, dass sie syngenetische Einschlüsse sind, die durch die sehr hohe Formenergie des umgebenden Diamanten morphologisch unter Zwang stehen. Aus zweidiemnsionalen Messungen der ersten Ordnung von charakteristischen Raman-Banden lassen sich relative Restdrucke in Diamanten zwischen Diamant und Einschlussmineral gewinnen; sie haben charakteristische Werte von ca. 0,4 bis 0,9 GPa um Chrom-Pyrop-Einschlüsse, 0,6 bis 2,0 GPa um Chrom-Diopsid-Einschlüsse, 0,3 bis 1,2 GPa um Olivin-Einschlüsse, 0,2 bis 1,0 GPa um Chromit-Einschlüsse, beziehungsweise 0,5 GPa um Graphit Einschlüsse.rnDie kristallstrukturellen Beziehung von Diamanten und ihren monomineralischen Einschlüssen wurden mit Hilfe der Quantifizierung der Winkelkorrelationen zwischen der [111] Richtung von Diamanten und spezifisch ausgewählten Richtungen ihrer mineralischen Einschlüsse untersucht. Die Winkelkorrelationen zwischen Diamant [111] und Chrom-Pyrop [111] oder Chromit [111] zeigen die kleinsten Verzerrungen von 2,2 bis zu 3,4. Die Chrom-Diopsid- und Olivin-Einschlüsse zeigen die Missorientierungswerte mit Diamant [111] bis zu 10,2 und 12,9 von Chrom-Diopsid [010] beziehungsweise Olivin [100].rnDie chemische Zusammensetzung von neun herausgearbeiteten (orientiertes Anschleifen) Einschlüssen (drei Chrom-Pyrop-Einschlüsse von Koffiefontein-, Finsch- und Venetia-Mine (zwei von drei koexistieren nebeneinander mit Enstatit), ein Chromit von Udachnaya (Sibirien/Russland), drei Chrom-Diopside von Koffiefontein, Koingnaas und Udachnaya (Sibirien/Russland) und zwei Olivin Einschlüsse von De Beers Pool und Koingnaas) wurden mit Hilfe EPMA und LA-ICP-MS analysiert. Auf der Grundlage der chemischen Zusammensetzung können die Mineraleinschlüsse in Diamanten in dieser Arbeit der peridotitischen Suite zugeordnet werden.rnDie Geothermobarometrie-Untersuchungen waren aufgrund der berührenden Koexistenz von Chrom-Pyrop- und Enstatit in einzelnen Diamanten möglich. Durchschnittliche Temperaturen und Drücke der Bildung sind mit ca. 1087 (± 15) C, 5,2 (± 0,1) GPa für Diamant DHK6.2 von der Koffiefontein Mine beziehungsweise ca. 1041 (± 5) C, 5,0 (± 0,1) GPa für Diamant DHF10.2 von der Finsch Mine zu interpretieren.rn

Periodicities in the Hodgkin-Huxley model and versions of this model with stochastic input

A field of computational neuroscience develops mathematical models to describe neuronal systems. The aim is to better understand the nervous system. Historically, the integrate-and-fire model, developed by Lapique in 1907, was the first model describing a neuron. In 1952 Hodgkin and Huxley [8] described the so called Hodgkin-Huxley model in the article “A Quantitative Description of Membrane Current and Its Application to Conduction and Excitation in Nerve”. The Hodgkin-Huxley model is one of the most successful and widely-used biological neuron models. Based on experimental data from the squid giant axon, Hodgkin and Huxley developed their mathematical model as a four-dimensional system of first-order ordinary differential equations. One of these equations characterizes the membrane potential as a process in time, whereas the other three equations depict the opening and closing state of sodium and potassium ion channels. The membrane potential is proportional to the sum of ionic current flowing across the membrane and an externally applied current. For various types of external input the membrane potential behaves differently. This thesis considers the following three types of input: (i) Rinzel and Miller [15] calculated an interval of amplitudes for a constant applied current, where the membrane potential is repetitively spiking; (ii) Aihara, Matsumoto and Ikegaya [1] said that dependent on the amplitude and the frequency of a periodic applied current the membrane potential responds periodically; (iii) Izhikevich [12] stated that brief pulses of positive and negative current with different amplitudes and frequencies can lead to a periodic response of the membrane potential. In chapter 1 the Hodgkin-Huxley model is introduced according to Izhikevich [12]. Besides the definition of the model, several biological and physiological notes are made, and further concepts are described by examples. Moreover, the numerical methods to solve the equations of the Hodgkin-Huxley model are presented which were used for the computer simulations in chapter 2 and chapter 3. In chapter 2 the statements for the three different inputs (i), (ii) and (iii) will be verified, and periodic behavior for the inputs (ii) and (iii) will be investigated. In chapter 3 the inputs are embedded in an Ornstein-Uhlenbeck process to see the influence of noise on the results of chapter 2.

EPR spectroscopy as a tool for the elucidation of non-covalent structuring principles in complex soft matter

This thesis aims at connecting structural and functional changes of complex soft matter systems due to external stimuli with non-covalent molecular interaction profiles. It addresses the problem of elucidating non-covalent forces as structuring principle of mainly polymer-based systems in solution. The structuring principles of a wide variety of complex soft matter types are analyzed. In many cases this is done by exploring conformational changes upon the exertion of external stimuli. The central question throughout this thesis is how a certain non-covalent interaction profile leads to solution condition-dependent structuring of a polymeric system.rnTo answer this question, electron paramagnetic resonance (EPR) spectroscopy is chosen as the main experimental method for the investigation of the structure principles of polymers. With EPR one detects only the local surroundings or environments of molecules that carry an unpaired electron. Non-covalent forces are normally effective on length scales of a few nanometers and below. Thus, EPR is excellently suited for their investigations. It allows for detection of interactions on length scales ranging from approx. 0.1 nm up to 10 nm. However, restriction to only one experimental technique likely leads to only incomplete pictures of complex systems. Therefore, the presented studies are frequently augmented with further experimental and computational methods in order to yield more comprehensive descriptions of the systems chosen for investigation.rnElectrostatic correlation effects in non-covalent interaction profiles as structuring principles in colloid-like ionic clusters and DNA condensation are investigated first. Building on this it is shown how electrostatic structuring principles can be combined with hydrophobic ones, at the example of host-guest interactions in so-called dendronized polymers (denpols).rnSubsequently, the focus is shifted from electrostatics in dendronized polymers to thermoresponsive alkylene oxide-based materials, whose structuring principles are based on hydrogen bonds and counteracting hydrophobic interactions. The collapse mechanism in dependence of hydrophilic-hydrophobic balance and topology of these polymers is elucidated. Complementarily the temperature-dependent phase behavior of elastin-like polypeptides (ELPs) is investigated. ELPs are the first (and so far only) class of compounds that is shown to feature a first-order inverse phase transition on nanoscopic length scales.rnFinally, this thesis addresses complex biological systems, namely intrinsically disordered proteins (IDPs). It is shown that the conformational space of the IDPs Osteopontin (OPN), a cytokine involved in metastasis of several kinds of cancer, and BASP1 (brain acid soluble protein one), a protein associated with neurite outgrowth, is governed by a subtle interplay between electrostatic forces, hydrophobic interaction, system entropy and hydrogen bonds. Such, IDPs can even sample cooperatively folded structures, which have so far only been associated with globular proteins.