Novel simulation methods for Coulomb and hydrodynamic interactions

This thesis presents new methods to simulate systems with hydrodynamic and electrostatic interactions. Part 1 is devoted to computer simulations of Brownian particles with hydrodynamic interactions. The main influence of the solvent on the dynamics of Brownian particles is that it mediates hydrodynamic interactions. In the method, this is simulated by numerical solution of the Navier--Stokes equation on a lattice. To this end, the Lattice--Boltzmann method is used, namely its D3Q19 version. This model is capable to simulate compressible flow. It gives us the advantage to treat dense systems, in particular away from thermal equilibrium. The Lattice--Boltzmann equation is coupled to the particles via a friction force. In addition to this force, acting on {it point} particles, we construct another coupling force, which comes from the pressure tensor. The coupling is purely local, i.~e. the algorithm scales linearly with the total number of particles. In order to be able to map the physical properties of the Lattice--Boltzmann fluid onto a Molecular Dynamics (MD) fluid, the case of an almost incompressible flow is considered. The Fluctuation--Dissipation theorem for the hybrid coupling is analyzed, and a geometric interpretation of the friction coefficient in terms of a Stokes radius is given. Part 2 is devoted to the simulation of charged particles. We present a novel method for obtaining Coulomb interactions as the potential of mean force between charges which are dynamically coupled to a local electromagnetic field. This algorithm scales linearly, too. We focus on the Molecular Dynamics version of the method and show that it is intimately related to the Car--Parrinello approach, while being equivalent to solving Maxwell's equations with freely adjustable speed of light. The Lagrangian formulation of the coupled particles--fields system is derived. The quasi--Hamiltonian dynamics of the system is studied in great detail. For implementation on the computer, the equations of motion are discretized with respect to both space and time. The discretization of the electromagnetic fields on a lattice, as well as the interpolation of the particle charges on the lattice is given. The algorithm is as local as possible: Only nearest neighbors sites of the lattice are interacting with a charged particle. Unphysical self--energies arise as a result of the lattice interpolation of charges, and are corrected by a subtraction scheme based on the exact lattice Green's function. The method allows easy parallelization using standard domain decomposition. Some benchmarking results of the algorithm are presented and discussed.

Computer simulation methods to study interfacial tensions : from the Ising model to colloidal crystals

In condensed matter systems, the interfacial tension plays a central role for a multitude of phenomena. It is the driving force for nucleation processes, determines the shape and structure of crystalline structures and is important for industrial applications. Despite its importance, the interfacial tension is hard to determine in experiments and also in computer simulations. While for liquid-vapor interfacial tensions there exist sophisticated simulation methods to compute the interfacial tension, current methods for solid-liquid interfaces produce unsatisfactory results.rnrnAs a first approach to this topic, the influence of the interfacial tension on nuclei is studied within the three-dimensional Ising model. This model is well suited because despite its simplicity, one can learn much about nucleation of crystalline nuclei. Below the so-called roughening temperature, nuclei in the Ising model are not spherical anymore but become cubic because of the anisotropy of the interfacial tension. This is similar to crystalline nuclei, which are in general not spherical but more like a convex polyhedron with flat facets on the surface. In this context, the problem of distinguishing between the two bulk phases in the vicinity of the diffuse droplet surface is addressed. A new definition is found which correctly determines the volume of a droplet in a given configuration if compared to the volume predicted by simple macroscopic assumptions.rnrnTo compute the interfacial tension of solid-liquid interfaces, a new Monte Carlo method called ensemble switch method'' is presented which allows to compute the interfacial tension of liquid-vapor interfaces as well as solid-liquid interfaces with great accuracy. In the past, the dependence of the interfacial tension on the finite size and shape of the simulation box has often been neglected although there is a nontrivial dependence on the box dimensions. As a consequence, one needs to systematically increase the box size and extrapolate to infinite volume in order to accurately predict the interfacial tension. Therefore, a thorough finite-size scaling analysis is established in this thesis. Logarithmic corrections to the finite-size scaling are motivated and identified, which are of leading order and therefore must not be neglected. The astounding feature of these logarithmic corrections is that they do not depend at all on the model under consideration. Using the ensemble switch method, the validity of a finite-size scaling ansatz containing the aforementioned logarithmic corrections is carefully tested and confirmed. Combining the finite-size scaling theory with the ensemble switch method, the interfacial tension of several model systems, ranging from the Ising model to colloidal systems, is computed with great accuracy.

Entwicklung massenspektrometrischer Isotopenverdünnungstechniken für die zuverlässige Bestimmung von Elementspuren und Schwefelspezies in Erdölprodukten

Einer der Hauptschwerpunkte der Arbeit lag in der Entwicklung einer spezies-spezifischen und einer spezies-unspezifischen GC-ICP-Q-MSIVA von Schwefelspezies in Petroprodukten. Es wurden hierzu Indikatoren, ausgehend von elementarem 34S-angereichertem Schwefel, im Mikromaßstab synthetisiert. Für die spezies-spezifische GC-ICP-Q-MSIVA wurde die erstmalige Synthese von 34S-markiertem Thiophen, Dibenzothiophen und 4-Methyldibenzothiophen verwirklicht. Als Indikatorsynthese für die spezies-unspezifische GC-ICP-Q-MSIVA erfolgte die erstmalige Darstellung von 34S-angereichertem Dimethyldisulid. Mit Hilfe der synthetisierten Verbindungen wurden spezies-spezifische und spezies-unspezifische massenspektrometrische Isotopenverdünnungsanalysen von Schwefelspezies in Petroprodukten durchgeführt. Vor allen GC-ICP-Q-MSIVA-Analysen erfolgte eine umfangreiche Speziesidentifizierung durch Aufstockexperimente mit kommerziell erhältlichen Standards und mit einem mit der GC gekoppelten Elektronenstoß (EI)-MS. Beide ICP-Q-MS Methoden zeichnen sich durch sehr niedrige Nachweisgrenzen (7 ng S/g) aus, welche auch eine Anwendbarkeit auf tiefentschwefelte Kraftstoffe garantieren. Mit der spezies-unspezifischen GC-ICP-Q-MSIVA ist neben einer Speziesanalyse auch eine Gesamtschwefelanalyse durch Aufsummierung aller in der Probe vorhandenen Spezies möglich. Es wurde im Rahmen dieser Arbeit auch der Einfluss möglicher Empfindlichkeitsänderungen des ICP-Q-MS durch koeluierende Kohlenwasserstoffe überprüft, wobei diese erwartungsgemäß auf das Ergebnis der spezies-spezifischen und spezies-unspezifischen GC-ICP-Q-MSIVA keinerlei Einfluss haben. Der zweite Hauptschwerpunkt der Arbeit lag auf der Ausarbeitung routinefähiger, schneller und zuverlässiger Methoden zur Gesamtelementspurenanalytik von Schwefel und Schwermetallen in Erdölen und Petroprodukten. Für die Gesamtschwefelanalyse wurde eine MSIVA nach thermaler Verdampfung mit 34S-markierten Dibenzothiophen als Indikator entwickelt. Die neu entwickelte Methode erlaubt eine sehr schnelle Bestimmung des Gesamtschwefelgehalts, wobei die eigentliche Messung des Isotopenverhältnisses innerhalb von Sekunden nach der Injektion der Probe erfolgt. Weiterhin zeichnet sich die Methode durch Robustheit und eine niedrige Nachweisgrenze (40 ng S/g) aus. Für die Analyse von Schwermetallen wurden erstmals Möglichkeiten einer direkten MSIVA von Erdölproben ohne zeitraubenden, kontaminationsträchtigen Aufschluss bzw. die schwierige Erzeugung einer Mikroemulsion zwischen hydrophober Probe und wässrigem Indikator entwickelt. Um eine homogene Verteilung des Indikators in der hydrophoben Probe zu ermöglichen, musste ausgehend von den zur Verfügung stehenden wässrigen Indikatorlösungen, eine Überführung des Indikators in ein organisches Lösungsmittel erfolgen. Hierzu wurde der jeweilige Metallindikator unter Komplexierung aus wässrigen Metallindikatorlösungen extrahiert. Für die Analyse der mit diesen Indikatorlösungen in organischer Phase versetzten Proben wurden zwei alternative Methoden ausgearbeitet. Bei der mit der Laserablation (LA) kombinierten ICP-SF-MSIVA wird die isotopenverdünnte Probe aus einer eigens für diesen Zweck entwickelten Probenhalterung ablatiert und so dem ICP-SF-MS zugeführt wird. Bei zeitlich sich verändernden Intensitäten der gemessenen Isotope werden aber reproduzierbare und konstante Isotopenverhältnisse erhalten. Im Falle einer homogenen Verteilung der Metallspuren wurde eine hervorragende Übereinstimmung mit Vergleichsmethoden und einem Referenzmaterial festgestellt. Im Falle einer heterogenen partikulären Verteilung der Metallspuren, wie sie z.B. bei Eisenspuren in den Erdölen vorlag, ist die Anwendbarkeit der LA-ICP-SF-MSIVA aufgrund des kleinen Probenvolumens (20 µL) jedoch begrenzt. Als Alternative zur LA-ICP-SF-MSIVA wurde ein System unter Verwendung der Fließinjektion für die Zuführung der isotopenverdünnten Probe zum ICP-SF-MS ausgearbeitet. Die isotopenverdünnte Probe wird hierbei in einen Eluentenstrom von Toluol injiziert und mit Hilfe einer Total-Consumption-Zerstäuber/Sprühkammer-Einheit vollständig bei einer Flussrate von 10 µL/min in das Plasma eingebracht. Neben einer nochmaligen Verkürzung der Analysenzeit und Vereinfachung der Probenvorbereitung bietet diese Methode zusätzlich stark verbesserte Nachweisgrenzen (z.B. Ni 0,9 ng/g). Leider sind mit diesem Verfahren bis jetzt nur Ni und Mo zuverlässig bestimmbar. Das in dieser Arbeit ausgearbeitete Methodenpaket erlaubt erstmals eine breite Einführung der ICP-MSIVA als zuverlässige Methode in die Routineanalytik der Petroindustrie. Durch die bewiesene Zuverlässigkeit, den geringen Zeitaufwand und die Robustheit der Methoden steht ihrem routinemäßigen Einsatz, außer einer weitergehenderen Automatisierung einzelner Verfahrensteile, prinzipiell nichts entgegen.

Finite-element discretization of 3D energy-transport equations for semiconductors

In this thesis a mathematical model was derived that describes the charge and energy transport in semiconductor devices like transistors. Moreover, numerical simulations of these physical processes are performed. In order to accomplish this, methods of theoretical physics, functional analysis, numerical mathematics and computer programming are applied. After an introduction to the status quo of semiconductor device simulation methods and a brief review of historical facts up to now, the attention is shifted to the construction of a model, which serves as the basis of the subsequent derivations in the thesis. Thereby the starting point is an important equation of the theory of dilute gases. From this equation the model equations are derived and specified by means of a series expansion method. This is done in a multi-stage derivation process, which is mainly taken from a scientific paper and which does not constitute the focus of this thesis. In the following phase we specify the mathematical setting and make precise the model assumptions. Thereby we make use of methods of functional analysis. Since the equations we deal with are coupled, we are concerned with a nonstandard problem. In contrary, the theory of scalar elliptic equations is established meanwhile. Subsequently, we are preoccupied with the numerical discretization of the equations. A special finite-element method is used for the discretization. This special approach has to be done in order to make the numerical results appropriate for practical application. By a series of transformations from the discrete model we derive a system of algebraic equations that are eligible for numerical evaluation. Using self-made computer programs we solve the equations to get approximate solutions. These programs are based on new and specialized iteration procedures that are developed and thoroughly tested within the frame of this research work. Due to their importance and their novel status, they are explained and demonstrated in detail. We compare these new iterations with a standard method that is complemented by a feature to fit in the current context. A further innovation is the computation of solutions in three-dimensional domains, which are still rare. Special attention is paid to applicability of the 3D simulation tools. The programs are designed to have justifiable working complexity. The simulation results of some models of contemporary semiconductor devices are shown and detailed comments on the results are given. Eventually, we make a prospect on future development and enhancements of the models and of the algorithms that we used.

A mesoscopic simulation method for electrolyte solutions

This thesis deals with the development of a novel simulation technique for macromolecules in electrolyte solutions, with the aim of a performance improvement over current molecular-dynamics based simulation methods. In solutions containing charged macromolecules and salt ions, it is the complex interplay of electrostatic interactions and hydrodynamics that determines the equilibrium and non-equilibrium behavior. However, the treatment of the solvent and dissolved ions makes up the major part of the computational effort. Thus an efficient modeling of both components is essential for the performance of a method. With the novel method we approach the solvent in a coarse-grained fashion and replace the explicit-ion description by a dynamic mean-field treatment. Hence we combine particle- and field-based descriptions in a hybrid method and thereby effectively solve the electrokinetic equations. The developed algorithm is tested extensively in terms of accuracy and performance, and suitable parameter sets are determined. As a first application we study charged polymer solutions (polyelectrolytes) in shear flow with focus on their viscoelastic properties. Here we also include semidilute solutions, which are computationally demanding. Secondly we study the electro-osmotic flow on superhydrophobic surfaces, where we perform a detailed comparison to theoretical predictions.

Ab-initio study of hydrogen technology materials for hydrogen storage and proton conduction

This dissertation deals with two specific aspects of a potential hydrogen-based energy economy, namely the problems of energy storage and energy conversion. In order to contribute to the solution of these problems, the structural and dynamical properties of two promising materials for hydrogen storage (lithium imide/amide) and proton conduction (poly[vinyl phosphonic acid]) are modeled on an atomistic scale by means of first principles molecular dynamics simulation methods.rnrnrnIn the case of the hydrogen storage system lithium amide/imide (LiNH_2/Li_2NH), the focus was on the interplay of structural features and nuclear quantum effects. For these calculations, Path-Integral Molecular Dynamics (PIMD) simulations were used. The structures of these materials at room temperature were elucidated; in collaboration with an experimental group, a very good agreement between calculated and experimental solid-state 1H-NMR chemical shifts was observed. Specifically, the structure of Li_2NH features a disordered arrangement of the Li lattice, which was not reported in previous studies. In addition, a persistent precession of the NH bonds was observed in our simulations. We provide evidence that this precession is the consequence of a toroid-shaped effective potential, in which the protons in the material are immersed. This potential is essentially flat along the torus azimuthal angle, which might lead to important quantum delocalization effects of the protons over the torus.rnrnOn the energy conversion side, the dynamics of protons in a proton conducting polymer (poly[vinyl phosphonic acid], PVPA) was studied by means of a steered ab-initio Molecular Dynamics approach applied on a simplified polymer model. The focus was put on understanding the microscopic proton transport mechanism in polymer membranes, and on characterizing the relevance of the local environment. This covers particularly the effect of water molecules, which participate in the hydrogen bonding network in the material. The results indicate that these water molecules are essential for the effectiveness of proton conduction. A water-mediated Grotthuss mechanism is identified as the main contributor to proton conduction, which agrees with the experimentally observed decay on conductivity for the same material in the absence of water molecules.rnrnThe gain in understanding the microscopic processes and structures present in this materials can help the development of new materials with improved properties, thus contributing to the solution of problems in the implementation of fuel cells.

Mixed finite-element methods for elliptic convection-dominated problems arising in semiconductor physics

In this work we develop and analyze an adaptive numerical scheme for simulating a class of macroscopic semiconductor models. At first the numerical modelling of semiconductors is reviewed in order to classify the Energy-Transport models for semiconductors that are later simulated in 2D. In this class of models the flow of charged particles, that are negatively charged electrons and so-called holes, which are quasi-particles of positive charge, as well as their energy distributions are described by a coupled system of nonlinear partial differential equations. A considerable difficulty in simulating these convection-dominated equations is posed by the nonlinear coupling as well as due to the fact that the local phenomena such as "hot electron effects" are only partially assessable through the given data. The primary variables that are used in the simulations are the particle density and the particle energy density. The user of these simulations is mostly interested in the current flow through parts of the domain boundary - the contacts. The numerical method considered here utilizes mixed finite-elements as trial functions for the discrete solution. The continuous discretization of the normal fluxes is the most important property of this discretization from the users perspective. It will be proven that under certain assumptions on the triangulation the particle density remains positive in the iterative solution algorithm. Connected to this result an a priori error estimate for the discrete solution of linear convection-diffusion equations is derived. The local charge transport phenomena will be resolved by an adaptive algorithm, which is based on a posteriori error estimators. At that stage a comparison of different estimations is performed. Additionally a method to effectively estimate the error in local quantities derived from the solution, so-called "functional outputs", is developed by transferring the dual weighted residual method to mixed finite elements. For a model problem we present how this method can deliver promising results even when standard error estimator fail completely to reduce the error in an iterative mesh refinement process.

Noble-metal nanoparticles produced with colloidal lithography: fabrication, optical properties and applications

In this work, metal nanoparticles produced by nanosphere lithography were studied in terms of their optical properties (in connection to their plasmon resonances), their potential application in sensing platforms - for thin layer sensing and bio-recognition events -, and for a particular case (the nanocrescents), for enhanced spectroscopy studies. The general preparation procedures introduced early in 2005 by Shumaker-Parry et al. to produce metallic nanocrescents were extended to give rise to more complex (isolated) structures, and also, by combining colloidal monolayer fabrication and plasma etching techniques, to arrays of them. The fabrication methods presented in this work were extended not only to new shapes or arrangements of particles, but included also a targeted surface tailoring of the substrates and the structures, using different thiol and silane compounds as linkers for further attachment of, i.e. polyelectrolyte layers, which allow for a controlled tailoring of their nanoenvironment. The optical properties of the nanocrescents were studied with conventional transmission spectroscopy; a simple multipole model was adapted to explain their behaviour qualitatively. In terms of applications, the results on thin film sensing using these particles show that the crescents present an interesting mode-dependent sensitivity and spatial extension. Parallel to this, the penetrations depths were modeled with two simplified schemes, obtaining good agreement with theory. The multiple modes of the particles with their characteristic decay lengths and sensitivities represent a major improvement for particle-sensing platforms compared to previous single resonance systems. The nanocrescents were also used to alter the emission properties of fluorophores placed close to them. In this work, green emitting dyes were placed at controlled distances from the structures and excited using a pulsed laser emitting in the near infrared. The fluorescence signal obtained in this manner should be connected to a two-photon processes triggered by these structures; obtaining first insight into plasmon-mediated enhancement phenomena. An even simpler and faster approach to produce plasmonic structures than that for the crescents was tested. Metallic nanodiscs and nanoellipses were produced by means of nanosphere lithography, extending a procedure reported in the literature to new shapes and optical properties. The optical properties of these particles were characterized by extinction spectroscopy and compared to results from the literature. Their major advantage is that they present a polarization-dependent response, like the nanocrescents, but are much simpler to fabricate, and the resonances can be tailored in the visible with relative ease. The sensing capabilities of the metallic nanodiscs were explored in the same manner as for the nanocrescents, meaning their response to thin layers and to bio-recognition events on their surface. The sensitivity of these nanostructures to thin films proved to be lower than that of the crescents, though in the same order of magnitude. Experimental information about the near field extension for the Au nanodiscs of different sizes was also extracted from these measurements. Further resonance-tailoring approaches based on electrochemical deposition of metals on the nanodiscs were explored, as a means of modifying plasmon resonances by changing surface properties of the nanoparticles. First results on these experiments would indicate that the deposition of Ag on Au on a submonolayer coverage level can lead to important blue-shifts in the resonances, which would open a simple way to tailor resonances by changing material properties in a local manner.

Phasenverhalten, Kristallisation und Morphologie ladungsstabilisierter kolloidaler Mischungen

Die vorliegende Arbeit möchte die Anwendbarkeit ladungsstabilisierter kolloidaler Systeme als Modellsysteme für fundamentale Fragen der Festkörperphysik und Thermodynamik auf binäre Mischungen erweitern. In diesem Kontext untersucht sie das Phasenverhalten und mit ihm im Zusammenhang stehende Eigenschaften von binären Mischungen ladungsstabilisierter, sphärischer kolloidaler Partikel in wässriger Suspension. Da das Verhalten hartkugelähnlicher Systeme durch hohe Fremdionenkonzentrationen bereits gut bekannt ist, konzentriert sich diese Arbeit auf sehr langreichweitig repulsive Systeme unter deionisierten Bedingungen. Neben etablierten Methoden der Mikroskopie und statischer Lichtstreuung zur Phasendiagrammsbestimmung wird auch die Beobachtung der zeitabhängigen Entwicklung des Schermoduls verwendet, um eine langsame Erstarrungskinetik zu studieren. Es werden insbesondere Mischungen aus Komponenten unterschiedlicher Größe und Ladung der Größenverhältnisse 0,9, 0,82, 0,57, 0,39 und 0,37 untersucht. Diese zeigen in dieser Reihenfolge Phasendiagramme mit spindelförmigem fluid/kristallinen Koexistenzbereich wie auch azeotrope und eutektische Phasendiagramme. Die Strukturuntersuchungen aus der statischen Lichtstreuung stehen in praktisch allen Fällen im Einklang mit ungeordneten bcc- Substitutionskristallen, was über Modelle zu Schermodulmessungen bestätigt wird. Für das spindelförmige System wird ein überraschend weiter Koexistenzbereich beobachtet, wie er nicht von der Theorie erwartet wird. Die Lage, aber nicht die Form des Solidus stimmt quantitativ mit Simulationsvorhersagen zu einkomponentigen Systemen überein. Für das eutektische System bei einem Radienverhältnis von 0,57 wird der Einfluss der Schwerkraft auf das Phasenverhalten und die Erstarrungskinetik untersucht. Die der Kristallisation der kleineren Majoritätskomponente vorgelagerte gravitativ unterstützte Entmischung begünstigt hier die Verfestigung. Beobachtet werden Morphologien, die aus anderen Systemen bekannt sind (Facetten, Dendriten), wie auch erstmals eine kollumnare eutektische Morphologie. Aus den Ergebnissen wird der erste umfassende Überblick über das Phasenverhalten deionisierter Mischungen ladungsstabilisierter, sphärischer Partikel erstellt, die eine Diskussion der Daten anderer Autoren und unserer Gruppe über fluid-fluider Phasenseparation und einem System mit oberem azeotropen Punkt mit einschließt. Die meisten metallspezifischen Phasendiagrammtypen können mit ladungsstabilisierten kolloidalen Partikeln reproduziert werden. Die langreichweitig wechselwirkenden Partikel zeigt eine wesentlich verbesserte substitutionelle Mischbarkeit im Vergleich mit Hartkugel- und Metallsystemen. Das Größenverhältnis der sphärischen Partikel nimmt dabei die bestimmende Rolle für den Phasendiagrammtyp ein.