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Dynamics and statistics of hydrodynamically interacting particles in laminar flows (2011)

Jochen Bammert

The subject of this thesis is the investigation of the dynamics and statistics of hydrodynamically interacting particles in low Reynolds number flows, which is discussed in three interrelated themes. The first theme focuses on polymer fractionation. With our basic model we explore the possibility to sort dumbbells with respect to their size using a two dimensional periodic potential. It turns out that the purely diffusive behavior of a dumbbell in this structured landscape is dominated by the ratio of two characteristic length scales, namely the wavelength of the potential l and the size of the dumbbell b. We explain why the diffusion constant in the potential plane shows a pronounced local maximum around l/b equal 3/2. Furthermore, the influence of the spring rigidity and the hydrodynamic interaction on the diffusive motion are examined as well as the dumbbell statistics. If the dumbbell is driven by an external flow through the periodic landscape two different kinds of motion occur: transport along a potential valley and a stair-like motion oblique to the trenches. In the latter case, the dumbbell jumps regularly to a neighboring valley which results in an effective deflection. The onset of the oblique movement as well as the deflection angle beta depend on the hydrodynamic interaction, on the ratio l/b, and on the Brownian motion of the beads. Especially the significant dependence of beta on l/b enables particle sorting. The results are published. The second theme deals with the Brownian dynamics in shear flows. Here, we investigate the correlations of particle fluctuations in order to characterize the direct interplay between thermal motion, hydrodynamic interactions, and non-uniform flows.With respect to the experimental implementation the particles are caught by harmonic potentials. First, we consider one trapped Brownian bead in linear shear and Poiseuille flows. The correlation functions of the particle’s position and velocity fluctuations are calculated analytically. The main result is the occurrence of shear-induced cross-correlations between orthogonal fluctuations in the shear plane which are asymmetric in time. Moreover, the positional probability distribution, P(r), of a single bead in both types of flow is determined. In Poiseuille flow, where no analytical solutions can be obtained, we use perturbation expansions to derive formulas for P(r) that are valuable for the analysis of experimental data. In the case of a linear shear flow, a connection between the static correlations and the distribution functions is derived which allows a consistency check between independent measurements. Considering a system with several Brownian particles it is obvious that hydrodynamic interactions influence the correlations. In order to investigate this effect, we calculate the positional correlation functions for a setup of two trapped Brownian beads which are exposed to a linear shear flow. As expected, the one-particle correlations change compared to the single particle case described above. They depend on the distance between the two beads. In addition, we find inter-particle correlations between orthogonal positional fluctuations of different particles. The structure of these new cross-correlations depends significantly on the relative orientation of the two beads in the shear flow. They can have zero, one, or two local extrema as a function of time. In collaboration with Prof. Wagner from Saarbrücken some of our predictions are already confirmed by experiments, where polystyrene beads are caught by optical traps and simultaneously exposed to linear shear flows in a special microfluidic device. The results are published and further investigations are in progress.The third theme concentrates on the rheology of colloidal suspensions. Our deterministic model system consists of Hookean dumbbells suspended in a confined Newtonian fluid under constant shear. We perform a numerical study using fluid particle dynamics simulations, where the effective viscosity of the suspension, eta, and the dumbbell statistics are determined. The investigations on the tumbling motion of a single dumbbell reveals that eta is influenced by three different contributions: the volume fraction occupied by the dumbbell, the hydrodynamic interaction between the beads, and elastic correlation effects. For a suspension of independent spheres we observe in our simulations that the viscosity, as a function of the volume fraction Phi, differs from the prediction of Einstein, Batchelor and Green if Phi becomes larger than 8%. Replacing the beads by dumbbells leads to an increase of eta , which depends significantly on the length of the springs connecting the two beads. The distribution function for the orientation angle of the dumbbells indicates the complex motion of the individual objects in the suspension, which may lead to the so-called elastic turbulence, as experimentally discovered by Groisman and Steinberg.

Holographic Investigation of Azobenzene-Containing Low-Molecular-Weight Compounds (2011)

Hubert Audorff

In the present thesis, holographic volume and surface relief gratings in azobenzene-containing low-molecular-weight compounds are investigated to obtain a broader understanding of this new class of material. Azobenzene chromophores undergo light-induced trans-cis-trans isomerization cycles leading to a reorientation of the long axis of the chromophores. If linearly polarized light is incident on the sample, these isomerizations result in a reorientation of this axis perpendicular to the light polarization. A holographic light grating, which can be formed by the interference of two coherent laser beams, leads to the inscription of a refractive-index modulation in the material. The azobenzene-containing low-molecular-weight compounds studied in this thesis consist of different building blocks: the core unit, the azobenzene chromophores with substituents, and the spacer and the linkage group between chromophore and core unit. These components can be used in a modular-design principle to synthesize a large library of low-molecular-weight compounds. Surprisingly, some of the investigated low-molecular-weight compounds form also a liquid-crystalline phase besides the amorphous phase as evidenced by polarized microscopy. If these liquid-crystalline compounds are prepared as solid films, however, they are quenched to an amorphous state. Upon reorientation of the azobenzene chromophores by illumination with a holographic light grating, a phase transition from the amorphous to an ordered state can be induced. This phase change in the latent liquid-crystalline low-molecular-weight compounds is very interesting for applications. The latent liquid-crystalline low-molecular-weight compounds show a post-development of the refractive-index modulation after the writing process. The holographic gratings are even stable at temperatures higher than the glass transition temperature, which further proves the light-induced formation of an ordered domain. In contrast to their polymeric counterparts, molecular materials are expected to show a faster response to light because of the absence of polymer chain entanglements. Therefore, molecular glasses can be used as blending material for photo-addressable polymers to improve the photo-sensitivity of the blend as compared to the pure polymer. The influence of the core and the substituent was investigated in low-molecular-weight compounds which do not form liquid-crystalline phases. An azobenzene-containing diblock copolymer for holographic data storage consists of an inert majority block and a minority block containing the covalently bound photo-sensitive azobenzene chromophores. Blending a few weight percent of the optimized molecular glass to the diblock copolymer leads to an increase of sensitivity with increasing content of the molecular glass, mainly because the writing time to the maximum of the refractive-index modulation decreased. The increase of the sensitivity is much larger than the observed rise of the refractive-index modulation due to the higher concentration of azobenzene chromophores. It was demonstrated that the shorter writing times are not caused by thermal effects, the molecules of the molecular glass in the inert block, or by changes of the free volume or the morphology, but that they are due to the azobenzene chromophores of the molecular glass in the minority block. They reorient faster than the chromophores attached to the polymer backbone and, thereby create free volume. Additionally, they can assist the reorientation of the azobenzene chromophores bound to the polymer by cooperative effects, i.e. dipolar and steric interactions. Both effects result in shorter writing time and higher sensitivity of the system. In a blend containing two weight percent of the molecular glass, the inscribed gratings are still long-term stable and the sensitivity increases by a factor of 1.7 as compared to the pure diblock copolymer. Upon illumination of an azobenzene-containing material with a holographic light grating, besides the volume grating, also a surface relief grating can develop. Surface modulations with heights of up to 600 nm were achieved in molecular glasses. It was found that the build-up of the surface relief grating depends on the electrical susceptibility of the material at the optical frequency of the laser and the polarization of the laser beams. These experimental findings are in agreement with the gradient force model. According to this theory, the macroscopic material transport results from the forces on the polarized material in the electrical field gradient caused by the holographic light grating. For many applications it is important that the holographically produced surface relief gratings can be transferred to polymer surfaces. Replica molding can be used to easily copy the surface modulations to e.g. polycarbonate.

Modulation of Incommensurately Modulated Structures Studied by the Maximum Entropy Method (2011)

Li Liang

In this thesis, modulated properties of incommensurately modulated crystals are studied by the Maximum Entropy Method (MEM). This was done exemplarily on the compounds Rubidium tetrachlorozincate and Chromium pyrophosphate. To do so, the MEM derived (3+1)-dimensional superspace electron density was analysed to receive information on atomic positions and their modulation, and structure refinement by the method of least-squares and interpretation of the difference-Fourier maps were performed to better describe the atomic displacement parameters (ADPs) and to improve the applied structural models. All the MEM calculations are done by the computer program BayMEM. The analysis of this MEM reconstructed density map is done by the computer program EDMA. Our results suggest that the modulation of harmonic ADPs, anharmonic ADPs and its modulation form an intrinsic part of incommensurately modulated structures. We have shown that with a data set of certain resolution and satellite reflections of higher order, the inclusion of modulation of harmonic ADPs, the anharmonic ADPs and the modulation of anharmonic ADPs can significantly improve the fit of the structure model to the diffraction data. Such model then better represents the true nature of the structure under investigation.

Der Einfluss der Ketten- und der Persistenzlänge auf den Soret-Effekt in verdünnten Polymerlösungen (2010)

Dominik Stadelmaier

In dieser Arbeit wurde der Einfluss der Kettenlänge und -steifheit auf den Soret-Effekt in verdünnten Polymerlösungen untersucht. Dazu wurden die Transportkoeffizienten Soret-Koeffizient, Thermodiffusionskoeffizient und Diffusionskoeffizient D an Polymerlösungen im Grenzfall unendlicher Verdünnung bei einer Temperatur von 22 °C bestimmt. Bei den untersuchten Polymeren handelt es sich um Polystyrol (PS), Polydimethylsiloxan (PDMS), Polymethylmethacrylat (PMMA) und Poly-t-butylmethacrylat (PtBMA). Zudem wurden die linearen Alkane Hexan, Oktan, Dekan, Dodekan, Pentadekan, Hexadekan und Eicosan gemessen, die alle als Oligomere des Polymers Polyethylen (PE) betrachtet werden können. Als Lösungsmittel wurden Chloroform, Cyclohexan, Cyclohexanon, Cyclooktan, Ethylacetat, Ethylbenzol, Methyl-Ethyl-Keton (MEK), Tetrahydrofuran (THF) und Toluol verwendet. Zur Untersuchung des Einflusses der Kettenlänge auf den Soret-Effekt wurden Experimente an PS-Oligomeren in einem Molmassenbereich von 10 kg/mol bis hin zum effektiven Monomer Ethylbenzol sowie an den genannten linearen Alkanen durchgeführt. Der Einfluss der Persistenzlänge wurde durch Messungen an hochpolymeren Lösungen der Polymere PDMS, PMMA und PtBMA untersucht, wobei zusätzlich die auf unendlich große Molmassen extrapolierten Werte der n-Alkane herangezogen wurden. Bei den Untersuchungen der PS-Oligomere wurde in dieser Arbeit festgestellt, dass bei großen Polymermassen die Lösungsmittelviskosität der dominierende und einzig signifikante Parameter des Lösungsmittels für Thermodiffusion wird und dass das Produkt eta D_T einen gemeinsamen Plateau-Wert erreicht. Bei kürzeren Ketten nehmen die Thermodiffusionskoeffizienten ab, zudem liegen die Datenpunkte für verschiedene Lösungsmittel nicht mehr auf einer gemeinsamen Kurve. In einigen Fällen wird sogar ein Vorzeichenwechsel des Soret-Koeffizienten beobachtet. Ein Vergleich der Ergebnisse zweier effektiver Monomere, die sich nur bezüglich ihrer Endgruppe unterscheiden, zeigt, dass die Abnahme des Thermodiffusionskoeffizienten bei Verkürzung der Kettenlänge nicht auf einen Endgruppeneffekt zurückzuführen ist. Auch die früher in der Literatur zu findende Annahme, dass die Molmassenunabhängigkeit bei großen Massen eine Monomereigenschaft des Polymers ist, wird durch diese Beobachtung nicht gestützt. Die für Thermodiffusion relevanten Einheiten sind stattdessen korrelierte Segmente mit einer dem Kuhnschen Segment vergleichbaren Größe. Der Soret-Koeffizient steigt bei kurzen Ketten mit zunehmender Masse monoton an. Ein Wechsel des Lösungsmittels führt zu einer nahezu konstanten Verschiebung, unabhängig von der Molmasse. Für lange Polymerketten wird der hydrodynamische Radius die dominierende Eigenschaft für den Soret-Koeffizienten. Die Messungen der linearen Alkane ergänzen die für PS entwickelte Beschreibung durch Hinzunahme eines Polymers mit signifikant kürzerem korrelierten Segment. Auch für das Polymer PE verhalten sich sehr kurze Oligomere wie kleine Moleküle, d.h. es existieren keine einfachen Regeln zur Vorhersage des thermophoretischen Verhaltens. Anders als bei PS besitzen alle gemessenen Alkanlösungen negative Thermodiffusionskoeffizienten. Der molmassenunabhängige Bereich ist messtechnisch nicht zugänglich, da längere PE-Ketten bei Raumtemperatur in praktisch allen Lösungsmitteln unlöslich sind. Eine Extrapolation der Messdaten auf unendlich große Molmassen ergibt, anders als bei PS, molmassenabhängige Plateaus für das Produkt eta D_T. Die Untersuchungen der hochpolymeren Polymerlösungen von PDMS, PMMA und PtBMA erweitern das empirische Modell um den Einfluss unterschiedlicher Persistenzlängen auf den Soret-Effekt. Bei hinreichend großen Einheiten, die der Thermodiffusion unterliegen, ergeben sich lösungsmittelunabhängige Plateau-Werte für eta D_T. Darüber hinaus wird der Thermodiffusionskoeffizient unabhängig von der chemischen Natur der Polymere. Voraussetzung dafür ist offenbar, dass die Masse der korrelierten Segmente um mindestens eine Größenordnung über der Masse des Lösungsmittels liegt. Für sehr flexible Polymere hingegen werden niedrigere Werte für eta D_T erreicht und die Plateau-Werte bleiben lösungsmittelabhängig. Auch negative Thermodiffusionskoeffizienten treten auf. Eine Extrapolation der konzentrationsabhängig gemessenen Thermodiffusionskoeffizienten von Literaturdaten zeigt, dass die meisten kleineren Moleküle den für Polymere typischen Plateau-Wert nicht erreichen. Eine mögliche Schlussfolgerung daraus ist, dass die hier formulierte empirische Beschreibung nicht nur für korrelierte Segmente innerhalb einer Polymerkette, sondern auch für Einzelmoleküle gültig ist. Allerdings existieren auch Systeme, bei denen Abweichungen von den genannten Gesetzmäßigkeiten auftreten.

Bogoliubov Excitations of Inhomogeneous Bose-Einstein Condensates (2010)

Christopher Gaul

In this thesis, different aspects of interacting ultracold bosons in presence of inhomogeneous external potentials are studied. The first part deals with repulsively interacting Bose-Einstein condensates in speckle disorder potentials. In the Bogoliubov approach, the many-body problem is split into the Gross-Pitaevskii condensate (mean-field) and the Bogoliubov excitations, which are bosonic quasiparticles. The disorder potential causes an imprint in the condensate, which makes the Hamiltonian for the Bogoliubov excitations inhomogeneous. The inhomogeneous Bogoliubov Hamiltonian is the starting point for a diagrammatic perturbation theory that leads to the renormalized Bogoliubov dispersion relation. From this effective dispersion relation, physical quantities are derived, e.g. the mean free path and disorder corrections to the speed of sound and the density of states. The analytical results are supported by a numerical integration of the Gross-Pitaevskii equation and by an exact diagonalization of the disordered Bogoliubov problem. In the second part, Bloch oscillations of Bose-Einstein condensates in presence of time-dependent interactions are considered. In general, the interaction leads to dephasing and destroys the Bloch oscillation. Feshbach resonances allow the atom-atom interaction to be manipulated as function of time. In particular, modulations around zero are considered. Different modulations lead to very different behavior: either the wave packet evolves periodically with time or it decays rapidly. The former is explained by a periodic time-reversal argument. The decay in the other cases can be described by a dynamical instability with respect to small perturbations, which are similar to the Bogoliubov excitations in the first part.

Surface Deformations of Magnetic Continua in Homogeneous Fields (2009)

Christian Gollwitzer

In this thesis, experiments with magnetic liquids and gels are presented. Ferrofluids are synthetically created suspensions of magnetic nanoparticles in a carrier liquid. By adding a gelator, such a ferrofluid can be turned into a ferrogel. The magnetic properties of these substances are similar to a usual paramagnet with the important difference, that the susceptibility of the former is higher by a factor of 10^3 to 10^6. By the application of a homogeneous field, a transformation of the shape of a magnetic sample can be induced. In this thesis, four experiments on the surface deformation in homogeneous magnetic fields are presented. Two geometric configurations are considered: a horizontally extended flat layer with a free surface as well as a spherical sample. In both cases, the application of a homogeneous magnetic field leads to changes of the shape of the free boundary. In the case of the spherical geometry, the sample is deformed into a prolate ellipsoid under the action of the field, the so called magnetodeformational effect. In case of the extended flat layer, an abrupt shape transition into a patterned state takes place, the normal field or Rosensweig instability. In contrast to the smooth deformation of the sphere, this is an instability, which breaks the translational symmetry, and the transition occurs at a certain threshold value of the magnetic induction. Each of the four experiments in this thesis is briefly summarized in the following paragraphs. Part I of the thesis considers ferrofluids. In chapter 2, the ideal geometry of an infinitely extended flat layer is intentionally reduced to a cylinder such that only a single spike in the centre exists, and the solution space becomes rotationally symmetric. This makes the problem very feasible for experimental methods and numerical simulations. Two measurement techniques are applied and compared to each other, namely an X-ray technique, where the surface deformation is extracted from radioscopic images, and a laser technique, which focuses a laser spot onto the surface. The experiments and the simulations, the latter performed in close cooperation with a group in Athens, show a convincing agreement within a few percent. It remains an open question, whether the result can be deduced in analytic form, however. In chapter 3, a highly viscous ferrofluid is utilized to study the nonlinear dynamics of the normal field instability at very low Reynolds numbers. The linear growth rate for the growth and decay of the pattern at small amplitudes is extracted from the measurements and compared with an existing theoretical model. In addition, the measurement technique provides the reconstruction of a fully nonlinear amplitude equation, which is qualitatively compared to model equations. These nonlinear amplitude equations can only describe the dynamics of the growth in the immediate vicinity of the critical point so far. For a quantitative comparison, there is a need for a model with an extended range of validity. Additionally, localized patterns are observed which arise spontaneously in the neighbourhood of the unstable solution branch, which have previously been observed with the help of an external disturbance Part II of the thesis deals with thermoreversible ferrogels. Chapter 4 studies the magnetodeformational effect. A ferrogel sphere is exposed to homogeneous magnetic field. When the field is applied suddenly, the sphere not only elongates in the direction of the field, but also vibrates about the new equilibrium. On a longer time scale, the deformation continuously increases due to the viscoelastic properties of the gel. Both phenomena can well be described by a harmonic oscillator model, where the spring constant changes with time. From the deformation parallel and perpendicular to the applied field, Poisson´s ratio can be calculated, which turns out to be close to the limit of incompressibility. The absolute values of the deformation are compared to recent theoretical models. The resulting deviation of about 10% is attributed to the viscoelastic properties of the ferrogel, which are not taken into account in the static models. In chapter 5, the normal field instability is realized for the first time with a ferrogel. A flat layer of a thermoreversible ferrogel is exposed to a homogeneous magnetic field at different temperatures, where the gel is viscoelastic. This is a consequence of the need for a very soft material, such that the growth of the pattern is not completely suppressed by the elastic forces. The magnetic field is periodically modulated in time, and the amplitude of the instability is measured, which is modulated with the same frequency. The comparison with rheological measurements reveals a scaling of the modulated amplitude with the complex viscosity of the ferrogel. A comparison with the theoretical model for a ferrogel is difficult due to the viscoelasticity of the gel.

Self-interaction and charge transfer in organic semiconductors (2009)

Thomas Körzdörfer

The fascinating properties of organic molecular semiconductors paved the way for a new class of electronic devices such as organic light-emitting diodes, transistors, or solar cells. Despite an inferior efficiency as compared to commonly used silicon-based technologies, organic semiconductors promise the advent of fully flexible devices for large-area displays and solar cells, printed transistors as low-cost radio frequency identification (RFID) tags, displays for electronic books, and disposable measuring instruments for medical diagnosis. Hence, the investigation of organic molecular semiconductors has emerged as a vibrant field of development both in industry and in academia, spanning a wide range of subjects from physics, chemistry, and materials science to engineering and technology. Theoretical physicists can contribute to this progress by developing methods that allow to determine the electronic properties of organic semiconductors from first principles and thus deepen our knowledge of the underlying electronic processes in organic electronic devices. The calculation of the electronic properties of molecular semiconductors issues a serious challenge to theoretical physicists and chemists. Typically, organic semiconductor molecules employ several hundreds of electrons. For systems of that size, approaches that work with model Hamiltonians are typically not accurate enough in predicting many important electronic properties. However, solving the many-particle Schrödinger-equation by employing highly accurate perturbation theory approaches is often numerically too expensive to be considered as a convenient alternative. Hence, density functional theory (DFT) naturally arises as the method of choice. However, although in theory DFT yields an exact formulation of quantum mechanics, the quality of the results obtained from DFT calculations in practice strongly depends on the used approximations to the so-called exchange-correlation functional. This work concentrates on the problem of self-interaction, which is one of the most serious problems of commonly used approximative density functionals. As a major result of this work, it is demonstrated that self-interaction plays a decisive role for the performance of different approximative functionals in predicting accurate electronic properties of organic molecular semiconductors. This is particularly true for the calculation of ionization potentials, photoelectron spectra, dissociation, and charge-transfer processes. In search for a solution to the self-interaction problem, a new concept for correcting commonly used density functionals for self-interaction is introduced and applied to a variety of systems, spanning small molecules, extended molecular chains, and organic molecular semiconductors. It is further shown that the performance of functionals that are not free from self-interaction can vary strongly for different systems and observables of interest, thus entailing the danger of misinterpretation of the results obtained from those functionals. The underlying reasons for the varying performance of commonly used density functionals are discussed thoroughly in this work. Finally, this thesis provides strategies that allow to analyze the reliability of commonly used approximations to the exchange-correlation functional for particular systems of interest. This cumulative dissertation is divided into three parts. Part I gives a short introduction into DFT and its time-dependent extension (TDDFT). Part II provides further insights into the self-interaction problem, presents a newly developed concept for the correction of self-interaction, gives an introduction into the publications, and discusses their basic results. Finally, the four publications on self-interaction and charge-transfer in extended molecular systems and organic molecular semiconductors are collected in Part III.

Characterization of phase transitions by the analysis of crystal structures (2009)

Joachim Angelkort

In this thesis results of the investigations of the mechanisms of solid-solid phase transitions are reported on basis of the exemplary characterization of the phase transition of the metalorganic compound Eu(SC36H49)2 and of the inorganic transition-metal compounds TiI3 and CrOCl. The phase transitions were surveyed temperature dependently by the performance of single-crystal X-ray diffraction experiments and measurements of the magnetic susceptibility. The X-ray diffraction experiments were carried out as data collections of integrated intensities of reflections and as measurements of profiles on selected reflections in so-called omega-2theta maps. The data sets of the integrated intensities were used to determine the crystal structures at different temperatures. By the comparison of the high- and the low-temperature crystal structures the mechanisms of the phase transitions of the compounds Eu(SC36H49)2 and TiI3 were determined. Furthermore the transition temperatures of all three compounds were determined by temperature-dependent measurements of intensities of superstructure reflections. From the omega-2theta maps the monoclinic lattice distortion of the low-temperature phase of CrOCl was determined.

A Study of Magnetic Helicity in Decaying and Forced 3D-MHD Turbulence (2009)

Shiva Kunar Malapaka

This thesis presents a numerical study of a property of three dimensional magnetohydrodynamic (3D-MHD) turbulence, namely, inverse cascade (spectral transport from small scales to large scales) of magnetic helicity. Magnetic helicity is defined as the volume integral of the dot product of the magnetic field and the magnetic vector potential. It characterizes the linkage and twists of the magnetic field lines. The inverse cascade is believed to be one of the causes of large-scale magnetic structure formation in the universe. This numerical studies is aimed at understanding how the inverse cascade of magnetic helicity effects other quantities of the turbulent flow. Two setups, namely, forced turbulence and decaying turbulence are studied. In the forced case, the numerical simulation setup consists of an initial energy distribution and a forcing localized in the small scales. The decaying setup consists of an initial energy distribution in the intermediate scales, which is allowed to decay naturally. The analysis of the results shows that several quantities in the turbulent flow, show self-similar behavior in their spectra, giving rise to power laws, which were hitherto unknown. Some of the quantities which are known to show power law behaviors exhibit different values to the power law exponents. These power law behaviors are analyzed together with the dimensional analysis of the eddy damped quasi normal Markovian (EDQNM) approximation equations, to attain a new relation which explains the evolution of large-scale magnetic structures in both the turbulent setups. The results are substantiated by the analysis of structure functions, probability density functions and correlation functions. Visualization of real space structures is also carried out. A mechanism to achieve large-scale magnetic structures from random small-scale magnetic fluctuations involving both the forced and decaying turbulences, is suggested.

Einfluss einer Scherströmung auf die thermischen Fluktuationen in einer Flüssigkeit (2009)

Lukas Holzer

Diese Dissertation beschäftigt sich im ersten Teil mit der Dynamik von Teilchen in einer Scherströmung und den durch die hydrodynamische Wechselwirkung zwischen Teilchen induzierten Effekten. Andererseits unterliegen kleine suspendierte Teilchen der Brownschen Bewegung, die durch hydrodynamische Fluktuationen des Lösungsmittels verursacht wird. Der Frage, wie sich diese hydrodynamischen Fluktuationen als Funktion der Scherrate von denjenigen in einer ruhenden Flüssigkeit unterscheiden, ist der Hauptteil der Arbeit gewidmet. Im ersten Teil werden als einfaches Modell für drei festgehaltene Polymere drei Kugeln in einer Scherströmung betrachtet, wobei jede Kugel in einem harmonischen Potential gefangen ist. Die Kugeln werden durch die Strömung aus ihren Ruhelagen ausgelenkt und oberhalb einer kritischen Scherrate und einer mittleren Geschwindigkeit gehen die über das Lösungsmittel wechselwirkenden Kugeln in eine oszillatorische Bewegung über. Im zweiten Teil der Arbeit werden die Fluktuationen des Geschwindigkeitsfeldes in einer Scherströmung mit Hilfe der um das Scherfeld linearisierten Navier-Stokes Gleichungen und der hydrodynamischen Fluktuationstheorie berechnet. Für die Korrelation unter den Geschwindigkeitsfluktuationen entlang der beiden zueinander orthogonalen Richtungen innerhalb der Scherebene ergeben sich gegenüber der ruhenden Flüssigkeit zusätzliche, von der Scherrate abhängige Beiträge. Diese Korrelation der Geschwindigkeiten an zwei unterschiedlichen Punkten r_1 und r_2, hängt auf komplexe Weise von der Orientierung des Verbindungsvektors r=r_1-r_2 ab und nimmt invers proportional mit dem Abstand ab: 1/r. Die Geschwindigkeitsfluktuationen der Flüssigkeit induzieren stochastische Kräfte auf ein suspendiertes Teilchen. Es sind diejenigen stochastischen Kräfte, die in der Langevin- Gleichung für das Teilchen Eingang finden. In einer ruhenden Flüssigkeit sind diese Kräfte in zwei zueinander orthogonalen Richtungen unkorreliert. In der vorliegenden Arbeit wird gezeigt, dass diese Kreuzkorrelation in einem Scherfluss endlich und in führender Ordnung proportional zur Scherrate ist. Die Korrelationen der Geschwindigkeitsfluktuationen wurden in einer Näherungsrechnung analytisch und unter Einbezug der Wände in einer ebenen Couette-Strömung numerisch berechnet. Die Ergebnisse aus diesen beiden Zugängen stimmen qualitativ überein. Für letzteren Fall konnte bereits in einer ruhenden Flüssigkeit eine Anisotropie der Verteilung der stochastischen Kräfte gefunden werden, wonach in engeren Kanälen die Kräfte parallel zu den Wänden verstärkt und diejenigen senkrecht dazu abgeschwächt werden.

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