Computer simulations of two-dimensional colloidal crystals under confinement and shear

In this thesis we are presenting a broadly based computer simulation study of two-dimensional colloidal crystals under different external conditions. In order to fully understand the phenomena which occur when the system is being compressed or when the walls are being sheared, it proved necessary to study also the basic motion of the particles and the diffusion processes which occur in the case without these external forces. In the first part of this thesis we investigate the structural transition in the number of rows which occurs when the crystal is being compressed by placing the structured walls closer together. Previous attempts to locate this transition were impeded by huge hysteresis effects. We were able to determine the transition point with higher precision by applying both the Schmid-Schilling thermodynamic integration method and the phase switch Monte Carlo method in order to determine the free energies. These simulations showed not only that the phase switch method can successfully be applied to systems with a few thousand particles and a soft crystalline structure with a superimposed pattern of defects, but also that this method is way more efficient than a thermodynamic integration when free energy differences are to be calculated. Additionally, the phase switch method enabled us to distinguish between several energetically very similar structures and to determine which one of them was actually stable. Another aspect considered in the first result chapter of this thesis is the ensemble inequivalence which can be observed when the structural transition is studied in the NpT and in the NVT ensemble. The second part of this work deals with the basic motion occurring in colloidal crystals confined by structured walls. Several cases are compared where the walls are placed in different positions, thereby introducing an incommensurability into the crystalline structure. Also the movement of the solitons, which are created in the course of the structural transition, is investigated. Furthermore, we will present results showing that not only the well-known mechanism of vacancies and interstitial particles leads to diffusion in our model system, but that also cooperative ring rotation phenomena occur. In this part and the following we applied Langevin dynamics simulations. In the last chapter of this work we will present results on the effect of shear on the colloidal crystal. The shear was implemented by moving the walls with constant velocity. We have observed shear banding and, depending on the shear velocity, that the inner part of the crystal breaks into several domains with different orientations. At very high shear velocities holes are created in the structure, which originate close to the walls, but also diffuse into the inner part of the crystal.

Numerical simulation of some viscoelastic fluids

Numerical simulation of the Oldroyd-B type viscoelastic fluids is a very challenging problem. rnThe well-known High Weissenberg Number Problem" has haunted the mathematicians, computer scientists, and rnengineers for more than 40 years. rnWhen the Weissenberg number, which represents the ratio of elasticity to viscosity, rnexceeds some limits, simulations done by standard methods break down exponentially fast in time. rnHowever, some approaches, such as the logarithm transformation technique can significantly improve rnthe limits of the Weissenberg number until which the simulations stay stable. rnrnWe should point out that the global existence of weak solutions for the Oldroyd-B model is still open. rnLet us note that in the evolution equation of the elastic stress tensor the terms describing diffusive rneffects are typically neglected in the modelling due to their smallness. However, when keeping rnthese diffusive terms in the constitutive law the global existence of weak solutions in two-space dimension rncan been shown. rnrnThis main part of the thesis is devoted to the stability study of the Oldroyd-B viscoelastic model. rnFirstly, we show that the free energy of the diffusive Oldroyd-B model as well as its rnlogarithm transformation are dissipative in time. rnFurther, we have developed free energy dissipative schemes based on the characteristic finite element and finite difference framework. rnIn addition, the global linear stability analysis of the diffusive Oldroyd-B model has also be discussed. rnThe next part of the thesis deals with the error estimates of the combined finite element rnand finite volume discretization of a special Oldroyd-B model which covers the limiting rncase of Weissenberg number going to infinity. Theoretical results are confirmed by a series of numerical rnexperiments, which are presented in the thesis, too.

Computer simulations of two-dimensional colloidal crystals in confinement

Monte Carlo simulations are used to study the effect of confinement on a crystal of point particles interacting with an inverse power law potential in d=2 dimensions. This system can describe colloidal particles at the air-water interface, a model system for experimental study of two-dimensional melting. It is shown that the state of the system (a strip of width D) depends very sensitively on the precise boundary conditions at the two ``walls'' providing the confinement. If one uses a corrugated boundary commensurate with the order of the bulk triangular crystalline structure, both orientational order and positional order is enhanced, and such surface-induced order persists near the boundaries also at temperatures where the system in the bulk is in its fluid state. However, using smooth repulsive boundaries as walls providing the confinement, only the orientational order is enhanced, but positional (quasi-) long range order is destroyed: The mean-square displacement of two particles n lattice parameters apart in the y-direction along the walls then crosses over from the logarithmic increase (characteristic for $d=2$) to a linear increase (characteristic for d=1). The strip then exhibits a vanishing shear modulus. These results are interpreted in terms of a phenomenological harmonic theory. Also the effect of incommensurability of the strip width D with the triangular lattice structure is discussed, and a comparison with surface effects on phase transitions in simple Ising- and XY-models is made

Factorization method and inclusions of mixed type in an inverse elliptic boundary value problem

In various imaging problems the task is to use the Cauchy data of the solutions to an elliptic boundary value problem to reconstruct the coefficients of the corresponding partial differential equation. Often the examined object has known background properties but is contaminated by inhomogeneities that cause perturbations of the coefficient functions. The factorization method of Kirsch provides a tool for locating such inclusions. In this paper, the factorization technique is studied in the framework of coercive elliptic partial differential equations of the divergence type: Earlier it has been demonstrated that the factorization algorithm can reconstruct the support of a strictly positive (or negative) definite perturbation of the leading order coefficient, or if that remains unperturbed, the support of a strictly positive (or negative) perturbation of the zeroth order coefficient. In this work we show that these two types of inhomogeneities can, in fact, be located simultaneously. Unlike in the earlier articles on the factorization method, our inclusions may have disconnected complements and we also weaken some other a priori assumptions of the method. Our theoretical findings are complemented by two-dimensional numerical experiments that are presented in the framework of the diffusion approximation of optical tomography.

Factorization method and irregular inclusions in electrical impedance tomography

In electrical impedance tomography, one tries to recover the conductivity inside a physical body from boundary measurements of current and voltage. In many practically important situations, the investigated object has known background conductivity but it is contaminated by inhomogeneities. The factorization method of Andreas Kirsch provides a tool for locating such inclusions. Earlier, it has been shown that under suitable regularity conditions positive (or negative) inhomogeneities can be characterized by the factorization technique if the conductivity or one of its higher normal derivatives jumps on the boundaries of the inclusions. In this work, we use a monotonicity argument to generalize these results: We show that the factorization method provides a characterization of an open inclusion (modulo its boundary) if each point inside the inhomogeneity has an open neighbourhood where the perturbation of the conductivity is strictly positive (or negative) definite. In particular, we do not assume any regularity of the inclusion boundary or set any conditions on the behaviour of the perturbed conductivity at the inclusion boundary. Our theoretical findings are verified by two-dimensional numerical experiments.

The development of stylolites, from small-scale heterogeneities to multi-scale roughness

Stylolites are rough paired surfaces, indicative of localized stress-induced dissolution under a non-hydrostatic state of stress, separated by a clay parting which is believed to be the residuum of the dissolved rock. These structures are the most frequent deformation pattern in monomineralic rocks and thus provide important information about low temperature deformation and mass transfer. The intriguing roughness of stylolites can be used to assess amount of volume loss and paleo-stress directions, and to infer the destabilizing processes during pressure solution. But there is little agreement on how stylolites form and why these localized pressure solution patterns develop their characteristic roughness.rnNatural bedding parallel and vertical stylolites were studied in this work to obtain a quantitative description of the stylolite roughness and understand the governing processes during their formation. Adapting scaling approaches based on fractal principles it is demonstrated that stylolites show two self affine scaling regimes with roughness exponents of 1.1 and 0.5 for small and large length scales separated by a crossover length at the millimeter scale. Analysis of stylolites from various depths proved that this crossover length is a function of the stress field during formation, as analytically predicted. For bedding parallel stylolites the crossover length is a function of the normal stress on the interface, but vertical stylolites show a clear in-plane anisotropy of the crossover length owing to the fact that the in-plane stresses (σ2 and σ3) are dissimilar. Therefore stylolite roughness contains a signature of the stress field during formation.rnTo address the origin of stylolite roughness a combined microstructural (SEM/EBSD) and numerical approach is employed. Microstructural investigations of natural stylolites in limestones reveal that heterogeneities initially present in the host rock (clay particles, quartz grains) are responsible for the formation of the distinctive stylolite roughness. A two-dimensional numerical model, i.e. a discrete linear elastic lattice spring model, is used to investigate the roughness evolving from an initially flat fluid filled interface induced by heterogeneities in the matrix. This model generates rough interfaces with the same scaling properties as natural stylolites. Furthermore two coinciding crossover phenomena in space and in time exist that separate length and timescales for which the roughening is either balanced by surface or elastic energies. The roughness and growth exponents are independent of the size, amount and the dissolution rate of the heterogeneities. This allows to conclude that the location of asperities is determined by a polimict multi-scale quenched noise, while the roughening process is governed by inherent processes i.e. the transition from a surface to an elastic energy dominated regime.rn

Flugzeuggetragene in-situ Messungen zur Eis- und Flüssigphase troposphärischer Wolken

Die Mikrophysik in Wolken bestimmt deren Strahlungseigenschaften und beeinflusst somit auch den Strahlungshaushalt des Planeten Erde. Aus diesem Grund werden im Rahmen der vorliegenden Arbeit die mikrophysikalischen Charakteristika von Cirrus-Wolken sowie von arktischen Grenzschicht-Wolken behandelt. Die Untersuchung dieser Wolken wurde mithilfe verschiedener Instrumente verwirklicht, welche Partikel in einem Durchmesserbereich von 250nm bis zu 6.4mm vermessen und an Forschungsflugzeugen montiert werden. Ein Instrumentenvergleich bestätigt, dass innerhalb der Bereiche in denen sich die Messungen dieser Instrumente überlappen, die auftretenden Diskrepanzen als sehr gering einzustufen sind. Das vorrangig verwendete Instrument trägt die Bezeichnung CCP (Cloud Combination Probe) und ist eine Kombination aus einem Instrument, das Wolkenpartikel anhand von vorwärts-gerichtetem Streulicht detektiert und einem weiteren, das zweidimensionale Schattenbilder einzelner Wolkenpartikel aufzeichnet. Die Untersuchung von Cirrus-Wolken erfolgt mittels Daten der AIRTOSS-ICE (AIRcraft TOwed Sensor Shuttle - Inhomogeneous Cirrus Experiment) Kampagne, welche im Jahr 2013 über der deutschen Nord- und Ostsee stattfand. Parameter wie Partikeldurchmesser, Partikelanzahlkonzentration, Partikelform, Eiswassergehalt, Wolkenhöhe und Wolkendicke der detektierten Cirrus-Wolken werden bestimmt und im Kontext des aktuellen Wissenstandes diskutiert. Des Weiteren wird eine beprobte Cirrus-Wolke im Detail analysiert, welche den typischen Entwicklungsprozess und die vertikale Struktur dieser Wolkengattung widerspiegelt. Arktische Grenzschicht-Wolken werden anhand von Daten untersucht, die während der VERDI (VERtical Distribution of Ice in Arctic Clouds) Kampagne im Jahr 2012 über der kanadischen Beaufortsee aufgezeichnet wurden. Diese Messkampagne fand im Frühling statt, um die Entwicklung von Eis-Wolken über Mischphasen-Wolken bis hin zu Flüssigwasser-Wolken zu beobachten. Unter bestimmten atmosphärischen Bedingungen tritt innerhalb von Mischphasen-Wolken der sogenannte Wegener-Bergeron-Findeisen Prozess auf, bei dem Flüssigwassertropfen zugunsten von Eispartikeln verdampfen. Es wird bestätigt, dass dieser Prozess anhand von mikrophysikalischen Messungen, insbesondere den daraus resultierenden Größenverteilungen, nachweisbar ist. Darüber hinaus wird eine arktische Flüssigwasser-Wolke im Detail untersucht, welche im Inneren das Auftreten von monomodalen Tröpfchen-Größenverteilungen zeigt. Mit zunehmender Höhe wachsen die Tropfen an und die Maxima der Größenverteilungen verschieben sich hin zu größeren Durchmessern. Dahingegen findet im oberen Übergangsbereich dieser Flüssigwasser-Wolke, zwischen Wolke und freier Atmosphäre, ein Wechsel von monomodalen zu bimodalen Tröpfchen-Größenverteilungen statt. Diese weisen eine Mode 1 mit einem Tropfendurchmesser von 20μm und eine Mode 2 mit einem Tropfendurchmesser von 10μm auf. Das dieses Phänomen eventuell typisch für arktische Flüssigwasser-Wolken ist, zeigen an dem Datensatz durchgeführte Analysen. Mögliche Entstehungsprozesse der zweiten Mode können durch Kondensation von Wasserdampf auf eingetragenen Aerosolpartikeln, die aus einer Luftschicht oberhalb der Wolke stammen oder durch Wirbel, welche trockene Luftmassen in die Wolke induzieren und Verdampfungsprozesse von Wolkentröpfchen hervorrufen, erklärt werden. Unter Verwendung einer direkten numerischen Simulation wird gezeigt, dass die Einmischung von trockenen Luftmassen in den Übergangsbereich der Wolke am wahrscheinlichsten die Ausbildung von Mode 2 verursacht.