Investigations of heterogeneous diffusion based on the probability density of scaled squared displacements observed from single molecules in ultra-thin liquid films

Diffusion processes in ultra-thin liquid films observed by video microscopy reveal a complex behavior. In contrast to homogeneous diffusion, dynamic and static heterogeneities are induced by layer transitions and compartments with differing diffusion coefficients, respectively. The objective of this research is the detection and distinction of such heterogeneities as well as an analysis of the underlying processes. Hence, a new method is proposed establishing a probability density of scaled squared displacements. This probability density allows for a simple and well-defined calculation of time-dependent diffusion coefficients and its fluctuations. Furthermore, by simulating a heterogeneous diffusion process these results are verified and compared to mean square displacement calculations. By means of the simulated probability density data, their dependency on the parameters is illustrated and further implications are pointed out

Single molecule tracking of the molecular mobility in thinning liquid films on thermally grown SiO 2

Diffusion coefficients obtained from weighted mean square displacements along probe molecule trajectories within ultrathin liquid TEHOS films show a correlation with film thickness. By studying cumulative distributions obtained with a time resolution of 20 ms, we could show that the diffusion is heterogeneous within our liquid films which consist of a few molecular layers only. We detected two components of the diffusion process, a slower and a faster one. Thinning of the film due to evaporation caused a slowdown of the whole diffusion process. But this resulted not from a slowdown in the two contributing components itself. Instead their relative contributions changed in favor for the slow component. We conclude that there is no pronounced difference in the diffusion coefficients attributed to the molecular layers 3 to 5 vertically above the substrate, but with the loss of upper layers along with the thinning process the concentration of probe molecules in the near surface region containing only one or two molecular layers is increased

Single molecule tracking of the molecular mobility in thinning liquid films on thermally grown SiO 2

Diffusion coefficients obtained from weighted mean square displacements along probe molecule trajectories within ultrathin liquid TEHOS films show a correlation with film thickness. By studying cumulative distributions obtained with a time resolution of 20 ms, we could show that the diffusion is heterogeneous within our liquid films which consist of a few molecular layers only. We detected two components of the diffusion process, a slower and a faster one. Thinning of the film due to evaporation caused a slowdown of the whole diffusion process. But this resulted not from a slowdown in the two contributing components itself. Instead their relative contributions changed in favor for the slow component. We conclude that there is no pronounced difference in the diffusion coefficients attributed to the molecular layers 3 to 5 vertically above the substrate, but with the loss of upper layers along with the thinning process the concentration of probe molecules in the near surface region containing only one or two molecular layers is increased

On the diffusion in inhomogeneous systems

Ziel dieser Arbeit ist die Untersuchung des Einflusses der stochastischen Interpretation der Langevin Gleichung mit zustandsabhängigen Diffusionskoeffizienten auf den Propagator des zugehörigen stochastischen Prozesses bzw. dessen Mittelwerte. Dies dient dem besseren Verständnis und der Interpretation von Messdaten von Diffusion in inhomogenen Systemen und geht einher mit der Frage der Form der Diffusionsgleichung in solchen Systemen. Zur Vereinfachung der Fragestellung werden in dieser Arbeit nur Systeme untersucht die vollständig durch einen ortsabhängigen Diffusionskoeffizienten und Angabe der stochastischen Interpretation beschrieben werden können. Dazu wird zunächst für mehrere experimentell relevante eindimensionale Systeme der jeweilige allgemeine Propagator bestimmt, der für jede denkbare stochastische Interpretation gültig ist. Der analytisch bestimmte Propagator wird dann für zwei exemplarisch ausgewählte stochastische Interpretationen, hier für die Itô und Klimontovich-Hänggi Interpretation, gegenübergestellt und die Unterschiede identifiziert. Für Mittelwert und Varianz der Prozesse werden die drei wesentlichen stochastischen Interpretationen verglichen, also Itô, Stratonovich und Klimontovich-Hänggi Interpretation. Diese systematische Untersuchung von inhomogenen Diffusionsprozessen kann zukünftig helfen diese Art von, in genau einer stochastischen Interpretation, driftfreien Systemen einfacher zu identifizieren. Ein weiterer wesentlicher Teil der Arbeit erweitert die Frage auf mehrdimensionale inhomogene anisotrope Systeme. Dies wird z.B. bei der Untersuchung von Diffusion in Flüssigkristallen mit inhomogenem Direktorfeld relevant. Obwohl hier, im Gegensatz zu eindimensionalen Systemen, der Propagator nicht allgemein berechnet werden kann, wird dennoch der Einfluss der Inhomogenität auf Messgrößen, wie die mittlere quadratische Verschiebung oder die Verteilung der Diffusivitäten, bestimmt. Anhand eines Beispiels wird auch der Einfluss der stochastischen Interpretation auf diese Messgrößen demonstriert.The aim of this thesis is to investigate the influence of the stochastic interpretation of the Langevin equation with state-dependent diffusion coefficient on the propagator of the related stochastic process, or its averages, respectively. This helps to obtain a deeper understanding and to interpret measurement data of diffusion in inhomogeneous systems and is accompanied with the question of the proper form of the diffusion equation in such systems. To simplify the question, in this thesis only systems are considered which can be fully described by a spatially dependent diffusion coefficient and a given stochastic interpretation. Therefore, for several experimentally relevant one-dimensional systems, the respective general propagator is determined, which is valid for any possible stochastic interpretation. Then, the propagator for two exemplary stochastic interpretations, here the Itô and Klimontovich-Hänggi interpretation, are compared and the differences are identified. For mean and variance of the processes three major interpretations are compared, namely the Itô, the Stratonovich and the Klimontovich-Hänggi interpretation. This systematic research on inhomogeneous diffusion process may help in future to identify these kind of, in exactly one stochastic interpretation, drift-free systems more easily. Another important part of this thesis extends this question to multidimensional inhomogeneous anisotropic systems. This is of high relevance, for instance, for the research of diffusion in liquid crystalline systems with an inhomogeneous director field. Although, in contrast to one-dimensional systems, the propagator may not be calculated generally, the influence of the inhomogeneity on measurement data like the mean squared displacement or the distribution of diffusivities is determined. Based on one example, also the influence of the stochastic interpretation on these quantities is demonstrated