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.

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.

New methods for the functionalization of metathesis polymers

Two general strategies for the functionalization of metathesis polymers are presented in this dissertation. Introducing Sacrificial Synthesis, many of the limitations of ruthenium-catalyzed ROMP have been overcome. Here, the living ROMP polymer to be functionalized was turned into a diblock copolymer by polymerizing dioxepine monomers onto the desired first polymer block. The second block was then later removed to leave “half-a-dioxepin”, i.e. exactly one hydroxyl group, at the chain-end. The efficiency of Sacrificial Synthesis is also studied. Thiol groups were also placed by a sacrificial strategy based on cyclic thioacetals. 2-Phenyl-1,3-dithiepin could be polymerized and subsequently cleaved by hydrogenation with Raney-Nickel. The presence of thiol groups on the chain end has been proven by chemical means (derivatization) and by coating gold-nanoparticles. The second strategy, vinyl lactone quenchingv is a termination reaction based on vinyl esters. After a metathesis step, an inactive Fischer-type carbene is formed. Such acyl carbenes are unstable and self-decompose to set an inactive ruthenium complex and the functional group free without changing the reaction conditions. The two compounds vinylene carbonate and 3H-furanone gave rise to the placement of aldehydes and carboxylic acids at the polymer chain ends without the necessity to perform any deprotection steps after the functionalization. The development of those two functionalization methods led to several applications. By reacting hydroxyl-functionalized ROMP-polymers with norbornene acid, macromonomers were formed which were subsequently polymerized to the respective graft-copolymers. Also, the derivatization of the same functionalized polymers with propargylic acid gave rise to alkyne-functionalized polymers which were conjugated with azides. Furthermore, “ugly stars”, i.e. long-chain branched structures were synthesized by polycondensation of ABn-type macromonomers and telechelic polymers were accessed combining the described functionalization techniques.