71 search hits
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The one-dimensional spin-1/2 ANNNI model in non-commuting magnetic fields
(2006)
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Adekunle Moses Adegoke
- In this thesis we have investigated the one-dimensional spin-1/2 Axial Next Nearest Neighbour Ising (ANNNI) model in non-commuting magnetic fields. As a starting point we obtained an estimate of the phase diagram of the model by treating the spins as classical vectors. This was followed by an investigation of the zero temperature ground state of the one-dimensional spin-1/2 ANNNI model in a longitudinal magnetic field. By using the symmetries of the Hamiltonian, we were able to diagonalize the longitudinal ANNNI model exactly. We found that there are four different possible ground state configurations for the longitudinal ANNNI model, in the thermodynamic limit. Rayleigh Schroedinger perturbation series for the ground state energy of the ANNNI model in non-commuting fields were then developed in each of the four ordered regions. Order parameters and the associated susceptibilities as well as specific heats were calculated. By application of the finite-size scaling technique it was possible to obtain the phase boundaries of the model numerically. For certain limits of the full Hamiltonian we compared the obtained results with the existing literature and we got good agreement.
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The Dynamics of Molecular Glasses Studied by Light Scattering
(2006)
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Sergei Adichtchev
- The present work is devoted to studying the dynamics in molecular glass formers applying the light-scattering (LS) technique, in particular tandem Fabry-Perot interferometry which allows to cover the frequency range from 0.3 GHz to 1000 GHz. Chapters 4 to 7 each present, in a self-contained way, different aspects of the dynamics, as summarized below. In Chapter 4, the results of the study of the molecular glass formers 2-picoline and m-tricresyl phosphate are presented. The LS spectra are analyzed in the frame of the Mode Coupling Theory (MCT). At high temperatures the evolution of the susceptibility minimum is well described by MCT. Below the critical temperature Tc, the asymptotic scaling laws of MCT fail due to the appearance of the excess wing of the alpha-process, which shows a universal evolution as a function of relaxation time, as was demonstrated by dielectric spectroscopy. A phenomenological approach, which allows to separate slow (alpha-process) and fast relaxation processes in the LS spectra is developed. Applying this approach, the temperature dependence of the non-ergodicity parameter f is obtained. The anomaly of f as well as a crossover to "white noise" of the fast dynamics spectra is found. In Chapter 5, the most extensive dielectric data of glycerol compiled by Lunkenheimer et al. [Contemp. Phys. 41, 15 (2000)] are reanalyzed. In contrast to the analysis of Lunkenheimer et al., the normalized susceptibility spectra, i.e, the dielectric loss data normalized by the static susceptibility, including the high temperature data, are analyzed. For this purpose a phenomenological approach, which describes the whole dielectric spectrum including the alpha-peak, its high frequency wing, and fast dynamics, is applied. The crossover temperature extracted from the phenomenological analysis and defined by the emergence of the high frequency wing upon cooling agrees well with the critical temperature extracted from the MCT analysis. The crossover temperature Tc=288 is significantly higher than reported before. Extracting the non-ergodicity parameter f, the characteristic anomaly similar to the one of 2-picoline and m-TCP discussed in Chapter 4, is found. In Chapter 6, the study of the fast relaxation below Tg in the molecular glasses 2-picoline, m-TCP, o-terphenyl (OTP), as well as in ethanol is presented. In addition to the boson peak, the depolarized LS spectra reveal quasi-elastic contributions that we attribute to i) the nearly constant loss (NCL) in the frequency range below 10 GHz and ii) a power law contribution with positive exponent alpha at higher frequencies. In the majority of glasses the latter may be attributed to thermally activated dynamics in asymmetric double well potentials (ADWP), as was previously found for the DLS spectra in silica. Following the Gilroy-Phillips model, the exponent alpha shows a master curve as a function of T /Vo for the various glasses where Vo specifies the width of the exponential distribution of barriers g(V), i.e., g(V) is propotional to the exp(-V/Vo). In Chapter 7, the investigation of the fast relaxation processes in the structural glass (T<Tg) and in the glassy crystal phase of ethanol, as well as in cyano cyclohexane, is presented. Depolarized and polarized LS spectra including the Brillouin lines were measured. It was found that depolarized, polarized LS and internal friction data exhibit fairly similar behavior, and thus reflect the same relaxations. The DLS spectra were described by assuming that the NCL contribution dominates below a few tens GHz, while the fast relaxational dynamics dominates at higher frequencies.
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Active and Passive Transport at Interfaces
(2011)
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Saeedeh Aliaskarisohi
- In this thesis we studied different forms of transport at interfaces. Four different interfacial transport mechanisms have been investigated. In each of them one physical aspect of active and passive transport is discussed. The four systems are arranged and discussed in four separate chapters. In chapter 3 and 4 we study the effect of static or hydrodynamic interactions on the cross over from individual diffusion towards collective diffusion. In chapter 3 the diffusion of circular domains on a giant unilamellar vesicle is measured. By tracking the motion of hydrodynamic interacting domains on a curved membrane we determined whether it is possible to extract rheological properties of the bilayer membrane. A similar two dimensional system interacting via static dipole interactions is studied in chapter 4. A mixture of paramagnetic and nonmagnetic colloidal particles immersed into a diluted ferrofluid is self assembled into colloidal flowers. In this experiment the effect of static interactions on the modes of diffusion of the petals of the colloidal flower is investigated in a one dimensional system. The results are compared with the single file diffusion of a hard core interacting one dimensional system. In chapter 5, the effect of actively directing particles with fluctuating active forces in a symmetry broken environment is studied. We address the question how to competing symmetry breaking effects decide on the direction of motion. The system consists of paramagnetic colloidal particles placed into an aqueous solution above the liquid-solid interface of a magnetic garnet film. An external modulated field supplies the fluctuations and the symmetry is broken by tilting the external field with respect to the magnetic film and/or by a magnetic symmetry broken pattern of the magnetic film. The direction of motion of the paramagnetic colloids is measured and we give a theoretical explanation of why which symmetry breaking wins. The fluidization of a two dimensional solid to a two dimensional liquid via the yielding of the monolayer is studied in chapter 6. The monolayer is locally yielded with thermo capillary interactions by focusing a laser onto it. We investigate the yielding as a function of the chemical nature of the monolayer and determine the thermodynamic requirements necessary for thermo capillary yielding.
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Characterization of phase transitions by the analysis of crystal structures
(2009)
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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.
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Holographic Investigation of Azobenzene-Containing Low-Molecular-Weight Compounds
(2011)
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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.
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Instabilities in layered liquids induced by external fields
(2003)
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Günter Auernhammer
- In this thesis, we have shown that the inclusion of a nematic degree of freedom in the macroscopic hydrodynamic description of smectic-A-like liquids leads to a number of interesting results. While the director and the layer normal are coupled such that they are parallel in equilibrium, in non-equilibrium situations, the director needs not be parallel to the smectic layer normal. This is in contrast to standard smectic-A hydrodynamics. Using irreversible thermodynamics and symmetry arguments, we derived a complete set of macroscopic hydrodynamic equations for the director variables, the layer displacement, the velocity field, and the moduli of the nematic and smectic order parameters. Recent experiments find that the parallel orientation of smectic-A- like liquids is destabilized by an applied shear. After destabilization, two typical scenarios are observed in a steady state situation: i) The layers are oriented perpendicular to the vorticity direction of the flow, i.e., they lie in the plane spanned by the velocity and the gradient direction (`perpendicular' orientation). ii) Closed multi-lamellar vesicles (`onions') form. A number of experiments indicate that the onset of this reorientation is controlled by the applied shear rate. In contrast to standard smectic-A hydrodynamics where shear in the parallel orientation has no effect on the layers, this destabilizing effect comes out naturally from our extended smectic-A hydrodynamics. The argumentation goes along the following lines. The shear field exerts a torque on the director that must be balanced by the coupling to the layer normal. In the limit of small angles, balancing these torques leads, in the steady state, to a shear-induced director tilt proportional to the shear rate. The preferred thickness of a smectic layer is directly connected to the projection of the averaged molecular axes on the layer normal, or, in terms of our model, the thickness is proportional to the projection of the director on the layer normal. If the director is tilted, this projection is shorter. This decrease of the projection is equivalent to an effective dilation, because the actual layer thickness is larger than the preferred layer thickness. Similar to the case of low molecular weight smectic-A liquid crystals under a dilative strain, this effective dilation leads to an undulation instability. To investigate the stability of the parallel alignment, we performed a linear and weakly non-linear analysis of the governing equations. The initial state is the above described spatially homogeneous director tilt with the smectic layers in the parallel orientation. The linear stability analysis showed an undulation instability which sets in above a critical tilt angle (or equivalently, a critical shear rate). This critical tilt angle turned out to depend strongly on the material parameters. For a typical low molecular weight thermotropic liquid crystal, we estimated the critical tilt angle to be on the order of a few degrees. The linear stability analysis also revealed that the nematic and smectic order is modulated close to the boundaries. Since the probability for the formation of defects is larger in regions with a decreased modulus of the order parameter, these variations in the modulus of the order parameter open the way for a destabilization of the layered structure. We note that a detailed investigation of this point is beyond the scope of the present work. Finally, we could exclude an oscillatory instability for all physically reasonable regions in parameter space. The weakly non-linear analysis shows that the bifurcation is supercritical for most physically relevant regions in the parameter space. A detailed comparison to an independent approach was undertaken in a collaboration with simulation physicists from the Max-Planck- Institute for Polymer Research in Mainz. In a molecular dynamics simulation, a model layered liquid consisting of chains of four particles (AABB) was considered. The interaction potential of particles not connected by springs is attractive for like particles and repulsive for particles of a different nature. The simulation demonstrated the two main predictions of our analytic theory: The director tilts in the flow direction and, above a critical shear rate, the layers show stationary undulations with a wave vector in the vorticity direction. Besides this good qualitative agreement, a reasonable quantitative agreement for the critical shear rate was found.
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Dynamics and statistics of hydrodynamically interacting particles in laminar flows
(2011)
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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.
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The Rosensweig instability in isotropic magnetic gels
(2008)
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Stefan Bohlius
- Die vorliegende Arbeit befasst sich mit der nichtlinearen theoretischen Analyse der Rosensweig Instabilität in isotropen magnetischen Gelen. Die Rosensweig Instabilität beschreibt den Übergang einer zunächst flachen Oberfläche zwischen einer magnetischen Flüssigkeit zu einer hexagonal geordneten Stacheloberfläche, sobald ein senkrecht zur flachen Oberfläche angelegtes homogenes Magnetfeld einen bestimmten kritischen Wert überschreitet. Startet man den Vernetzungsprozess in einer Mischung aus Polymeren, Vernetzungsreagenzien und einem Ferrofluid, so erhält man ein isotropes Ferrogel, ein elastisches Medium, welches zusätzlich superparamagnetisches Verhalten aufweist. Theoretisch lässt sich zeigen, dass auch die Oberfläche dieser Medien in einem angelegten Magnetfeld instabil wird, wobei die typische Wellenlänge im Vergleich zu gewöhnlichen Ferrofluiden unverändert bleibt, während die kritische Magnetfeldstärke mit wachsendem elastischen Schermodul steigt. Besondere Aufmerksamkeit kommt in der Diskussion dem stationären Charakter der Rosensweig Instabilität zu. Dieser ist, wie sich herausstellt, als ein Grenzprozess zu interpretieren, bei welchem die Dynamik der charakteristischen Mode mit Annäherung an die Schwelle immer stärker verlangsamt wird und schließlich zu einem statischen Oberflächenmuster führt. Der Grund für dieses Grenzverhalten ist in der deformierbaren Oberfläche und im Besonderen in der daraus resultierenden kinematischen Randbedingung zu sehen. Unter Anwendung der Energiemethode nach Gailitis, wird die Oberflächenenergiedichte bezüglich regulärer Streifen, Quadrate und Hexagone minimiert. Es zeigt sich, dass am Einsatz der Instabilität Hexagone das energetisch favorisierte Oberflächenmuster sind. Für hohe Magnetfeldstärken hingegen bilden Quadrate die bevorzugte Anordnung. Die Energiemethode hat jedoch bedeutende Nachteile, die als Motivation für eine schwach nichtlineare Analyse der fundamentalen hydrodynamischen Gleichungen und der Herleitung einer Amplitudengleichung dienen. Ganz besondere Beachtung verdient dabei die Bestimmung des adjungierten Systems für die Rosensweig Instabilität. Dieses ist zur Befriedigung der Fredholmschen Alternative, die wiederum die Amplitudengleichungen liefert, von zentraler Bedeutung. Zur Herleitung der adjungierten Gleichungen und der dazugehörigen Randbedingungen wird die Erkenntnis aus der Diskussion der linearen Instabilität, dass das System als dynamisch zu betrachten und der statische Grenzfall erst am Ende zu vollziehen ist, benutzt. Des weiteren stellt es sich als wichtig heraus, die Gleichungen zunächst für ein kompressibles Medium zu adjungieren und ebenfalls erst am Ende die Näherung für inkompressible Medien zu bestimmen. Das adjungierte System wird ebenfalls für die Marangoni Instabilität bestimmt. Dort induzieren Temperaturfluktuationen an der Oberfläche eines Fluids Fluktuationen der Oberflächenspannung, die wiederum Konvektion hervorrufen. Mit Hilfe der Lösungen des adjungierten Systems lassen sich nun die Lösbarkeitsbedingungen in der zweiten und dritten Störungsordnung erfüllen und man erhält letztlich die Amplitudengleichung. Im Rahmen unser Näherungen entkoppeln die hydrodynamischen Volumengleichungen von denen des Magnetfeldes. Allerdings müssen die Lösungen auch noch den Randbedingungen genügen und im Besonderen ist die normale Randbedingungen in den höheren Ordnungen nicht trivial erfüllt. Vielmehr liefert sie noch eine zusätzliche Bedingung zur Fredholmschen Alternative. In der Arbeit wird zum ersten Mal der quadratische Koeffizient aus den fundamentalen hydrodynamischen Gleichungen abgeleitet. Dieser garantiert zum einen die Existenz von Hexagonen, zum anderen das Auftreten einer transkritischen Bifurkation. Beides sind experimentell bestätigte Eigenschaften der Rosensweig Instabilität. Zum anderen enthält die Amplitudengleichung für Ferrogele eine zweifache Zeitableitung. Die linearisierte Amplitudengleichung nimmt im Fall der Ferrogele die Gestalt eines gedämpften harmonischen Oszillators an. Im Fall der Rosensweig Instabilität in Ferroflüssigkeiten, deren zugehörige Amplitudengleichung ebenfalls bestimmt wird, tritt diese zweifache Zeitableitung nicht auf. Die Rosensweig Instabilität ist im Rahmen unserer Näherungen rein oberflächengetriebenen. Das motiviert die Frage, inwieweit dünne magnetische Filme oder Membranen instabil werden können. Diese Frage wird in dieser Arbeit ebenfalls diskutiert. Beschränkt man sich in einer linearen Stabilitätsanalyse auf den symmetrischen Fall, das heißt der isotrope Ferrogelfilm ist auf beiden Seiten vom gleichen Medium umgeben, so findet man, dass der Film linear nicht instabil werden kann. Eine Instabilität zeigt sich nur im Fall von anisotropen magnetischen Gelen oder im Fall eines magnetischen Kontrastes zwischen den beiden umgebenden Medien.
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Thermische und langreichweitige Effekte in nichtlinearer Gitterdynamik mit Anwendungen in der Bio- und Nanophysik
(2006)
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Christian Brunhuber
- Der Gegenstand dieser Arbeit ist die Rolle von langreichweitigen Wechselwirkungen und thermischen Effekten in nichtlinearen Gittern und ihre Anwendung in der Bio- und Nanophysik. Thematisch unterteilt sich die Arbeit in drei Bereiche, in denen die bekannte Fermi-Pasta-Ulam-Kette, die diskrete nichtlineare Schroedingergleichung und zweidimensionale Fermi-Pasta-Ulam-Ebenen mit Graphit-aehnlichen Wechselwirkungspotenzialen der Teilchen untersucht werden. Im ersten Teil werden fruehere Studien zur thermischen Diffusion von nicht-topologischen Solitonen auf Fermi-Pasta-Ulam-Ketten mit langreichweitigen Wechselwirkungen erweitert. Zwei Typen von langreichweitigen Wechselwirkungen werden betrachtet: harmonische langreichweitige Wechselwirkungen mit Kac-Baker- und potenzartiger Abhaengigkeit der Kopplung vom Abstand der wechselwirkenden Teilchen. Das beobachtete superdiffusive Verhalten ist bestimmend fuer die Solitonen-Diffusion auf Ketten mit langreichweitigen Wechselwirkungen, wohingegen sie weniger ausschlaggebend fuer Ketten mit Naechste-Nachbar-Wechselwirkungen sind. Mit Hilfe einer Kollektiv-Variablen-Theorie und einem Variationsverfahren im Kontinuumslimit der Kette wird ein Langevin-System fuer die beiden kollektiven Variablen inverse Solitonenbreite und Solitonenposition aufgestellt. Simulationen und eine stoerungstheoretische Behandlung des Langevin-Systems zeigen, dass fuer beide Typen von langreichweitigen Wechselwirkungen die Solitonen (auf verschiedenen Zeitskalen) das gleiche Langzeitverhalten mit einer charakteristischen superdiffusiven Zeitabhaengigkeit in der Form eines Potenzgesetzes mit der Potenz 3/2 fuer die Positionsvarianz erreichen. Im zweiten Teil werden langreichweitige Wechselwirkungen vom Kac-Baker-Typ in die nichtlineare Schroedingergleichung mit und ohne Daempfung eingefuehrt. Die Kombination von langreichweitigen Kraeften und Daempfung fuehrt zu periodischen Strukturen von stationaeren diskreten Breathern die sich aus einem anfangs homogenen Untergrund entwickeln. Der Wechselwirkungsradius bestimmt die Periodizitaet, was in der Quasikontinuumsnaeherung des Systems verstanden werden kann. Fuer das ungedaempfte System steht der Einfluss der langreichweitigen Wechselwirkungen beim Uebergang in eine persistente Breather-Phase im Mittelpunkt, die nur von der Energie und der Norm des Systems abhaengt. Mit Hilfe von Monte-Carlo-Simulationen wird die Lokalisierungsstaerke als Funktion des Wechselwirkungsradius und der Systemtemperatur dargestellt, die formal negative Werte im Bereich der persistenten Breather-Phase annimmt. Der dritte Teil ist motiviert durch die Anwendung von Kohlenstoff-Nanoroehrchen in neuartigen Schaltungen. Die Kohlenstoff-Nanoroehrchen werden durch Fermi-Pasta-Ulam-Ebenen mit periodischen Randbedingungen in transversaler Richtung zur Achse des Roehrchens modelliert. In axialer Richtung werden zwei Waermebaeder angeschlossen. In den Simulationen wird der mittlere Waeremstrom und die thermische Leitfaehigkeit in Abhaengigkeit von der Laenge des Roehrchens und der mittleren Temperatur bestimmt. Die beobachteten Werte fuer die thermische Leitfaehigkeit stimmen mit frueheren Resultaten aus Molekulardynamik-Simulationen ueberein. Es hat sich gezeigt, dass die thermische Leitfaehigkeit einer zehn Atomabstaende breiten Fermi-Pasta-Ulam-Carbon-Ebene und einer Fermi-Pasta-Ulam-Kette fuer genuegend grosse Systeme mit der gleichen Potenz der Systemlaenge divergiert, die wiederum von der mittleren Temperatur abhaengt.
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Lagrangesche statistische Eigenschaften hydrodynamischer und magnetohydrodynamischer Turbulenz
(2009)
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Angela Busse
- In dieser Arbeit werden die Lagrangeschen statistischen Eigenschaften inkompressibler hydrodynamischer und magnetohydrodynamischer Turbulenz mit Hilfe direkter numerischer Simulationen untersucht. Magnetohydrodynamische Turbulenz wird hierbei in mehreren Konfigurationen, als makroskopisch isotrope Turbulenz, unter dem Einfluss eines starken mittleren Magnetfelds und im zweidimensionalen Fall, behandelt. Die Lagrangesche Statistik ist besonders zur Untersuchung des diffusiven und dispersiven Charakters turbulenter Strömungen geeignet. Im hydrodynamischen und magnetohydrodynamischen Fall treten in der turbulenten relativen Dispersion charakteristische Unterschiede auf. Es wird gezeigt, dass diese Unterschiede über das wechselnde Vorherrschen verschiedener Ausrichtungsprozesse begründet werden können. Diese Interpretation kann auch auf magnetohydrodynamische Turbulenz unter dem Einfluss eines starken mittleren Magnetfelds erweitert werden. Die Frage der phänomenologischen Beschreibung der turbulenten Energiekaskade ist im magnetohydrodynamischen Fall noch nicht gelöst. Im Eulerschen Bezugssystem ist es schwierig, zwischen verschiedenen Kaskadenmechanismen zu unterscheiden. Aus der Skalierung der Lagrangeschen Frequenzspektren ergeben sich Einsichten in die Zeitskalen und die Interaktionsmechanismen der turbulenten Energiekaskade. Die Frage der phänomenologischen Beschreibung der turbulenten Energiekaskade ist im magnetohydrodynamischen Fall noch nicht gelöst. Im Eulerschen Bezugssystem ist es schwierig, zwischen verschiedenen Kaskadenmechanismen zu unterscheiden. Aus der Skalierung der Lagrangeschen Frequenzspektren ergeben sich Einsichten in die Zeitskalen und die Interaktionsmechanismen der turbulenten Energiekaskade. Die Formen der Trajektorien der Testteilchen werden unter Berücksichtigung der Statistik der Krümmung und Beschleunigung im hydrodynamischen und magnetohydrodynamischen Fall verglichen. Weiterhin werden in dieser Arbeit verschiedene Antriebsverfahren, die in Simulationen hydrodynamischer Turbulenz verwendet werden, untersucht und auf den MHD-Fall erweitert.
