A molecular dynamics study of the influence of chain branching on the properties of polymer systems

Die vorliegende Arbeit beschäftigt sich mit dem Einfluß von Kettenverzweigungen unterschiedlicher Topologien auf die statischen Eigenschaften von Polymeren. Diese Untersuchungen werden mit Hilfe von Monte-Carlo- und Molekular-Dynamik-Simulationen durchgeführt.Zunächst werden einige theoretische Konzepte und Modelle eingeführt, welche die Beschreibung von Polymerketten auf mesoskopischen Längenskalen gestatten. Es werden wichtige Bestimmungsgrößen eingeführt und erläutert, welche zur quantitativen Charakterisierung von Verzweigungsstrukturen bei Polymeren geeignet sind. Es wird ebenso auf die verwendeten Optimierungstechniken eingegangen, die bei der Implementierung des Computerprogrammes Verwendung fanden. Untersucht werden neben linearen Polymerketten unterschiedliche Topolgien -Sternpolymere mit variabler Armzahl, Übergang von Sternpolymeren zu linearen Polymeren, Ketten mit variabler Zahl von Seitenketten, reguläre Dendrimere und hyperverzweigte Strukturen - in Abhängigkeit von der Lösungsmittelqualität. Es wird zunächst eine gründliche Analyse des verwendeten Simulationsmodells an sehr langen linearen Einzelketten vorgenommen. Die Skalierungseigenschaften der linearen Ketten werden untersucht in dem gesamten Lösungsmittelbereich vom guten Lösungsmittel bis hin zu weitgehend kollabierten Ketten im schlechten Lösungsmittel. Ein wichtiges Ergebnis dieser Arbeit ist die Bestätigung der Korrekturen zum Skalenverhalten des hydrodynamischen Radius Rh. Dieses Ergebnis war möglich aufgrund der großen gewählten Kettenlängen und der hohen Qualität der erhaltenen Daten in dieser Arbeit, insbesondere bei den linearen ketten, und es steht im Widerspruch zu vielen bisherigen Simulations-Studien und experimentellen Arbeiten. Diese Korrekturen zum Skalenverhalten wurden nicht nur für die linearen Ketten, sondern auch für Sternpolymere mit unterchiedlicher Armzahl gezeigt. Für lineare Ketten wird der Einfluß von Polydispersität untersucht.Es wird gezeigt, daß eine eindeutige Abbildung von Längenskalen zwischen Simulationsmodell und Experiment nicht möglich ist, da die zu diesem Zweck verwendete dimensionslose Größe eine zu schwache Abhängigkeit von der Polymerisation der Ketten besitzt. Ein Vergleich von Simulationsdaten mit industriellem Low-Density-Polyäthylen(LDPE) zeigt, daß LDPE in Form von stark verzweigten Ketten vorliegt.Für reguläre Dendrimere konnte ein hochgradiges Zurückfalten der Arme in die innere Kernregion nachgewiesen werden.

Partial reconstruction of the trajectories of a discretely observed branching diffusion with immigration and an application to inference

In this treatise we consider finite systems of branching particles where the particles move independently of each other according to d-dimensional diffusions. Particles are killed at a position dependent rate, leaving at their death position a random number of descendants according to a position dependent reproduction law. In addition particles immigrate at constant rate (one immigrant per immigration time). A process with above properties is called a branching diffusion withimmigration (BDI). In the first part we present the model in detail and discuss the properties of the BDI under our basic assumptions. In the second part we consider the problem of reconstruction of the trajectory of a BDI from discrete observations. We observe positions of the particles at discrete times; in particular we assume that we have no information about the pedigree of the particles. A natural question arises if we want to apply statistical procedures on the discrete observations: How can we find couples of particle positions which belong to the same particle? We give an easy to implement 'reconstruction scheme' which allows us to redraw or 'reconstruct' parts of the trajectory of the BDI with high accuracy. Moreover asymptotically the whole path can be reconstructed. Further we present simulations which show that our partial reconstruction rule is tractable in practice. In the third part we study how the partial reconstruction rule fits into statistical applications. As an extensive example we present a nonparametric estimator for the diffusion coefficient of a BDI where the particles move according to one-dimensional diffusions. This estimator is based on the Nadaraya-Watson estimator for the diffusion coefficient of one-dimensional diffusions and it uses the partial reconstruction rule developed in the second part above. We are able to prove a rate of convergence of this estimator and finally we present simulations which show that the estimator works well even if we leave our set of assumptions.

Mutually catalytic branching at infinite rate

The purpose of this doctoral thesis is to prove existence for a mutually catalytic random walk with infinite branching rate on countably many sites. The process is defined as a weak limit of an approximating family of processes. An approximating process is constructed by adding jumps to a deterministic migration on an equidistant time grid. As law of jumps we need to choose the invariant probability measure of the mutually catalytic random walk with a finite branching rate in the recurrent regime. This model was introduced by Dawson and Perkins (1998) and this thesis relies heavily on their work. Due to the properties of this invariant distribution, which is in fact the exit distribution of planar Brownian motion from the first quadrant, it is possible to establish a martingale problem for the weak limit of any convergent sequence of approximating processes. We can prove a duality relation for the solution to the mentioned martingale problem, which goes back to Mytnik (1996) in the case of finite rate branching, and this duality gives rise to weak uniqueness for the solution to the martingale problem. Using standard arguments we can show that this solution is in fact a Feller process and it has the strong Markov property. For the case of only one site we prove that the model we have constructed is the limit of finite rate mutually catalytic branching processes as the branching rate approaches infinity. Therefore, it seems naturalto refer to the above model as an infinite rate branching process. However, a result for convergence on infinitely many sites remains open.

From amphiphilic block copolymers to ferrocenyl-functionalized polymers for biosensoric applications

The present thesis can be divided in three main parts. In all parts new polymer architecturesrnwere synthesized and characterized concerning their special features.rnThe first part will emphasize the advantage of a polystyrene-block-(hyperbranchedrnpolyglycerol) copolymer in comparison to an analogue polystyrene-block-(linear polyglycerol)rncopolymer. Therefore a synthethic route to prepare linear block copolymersrnhas been developed. Two strategies were examined. One strategy was based on thernclassic, sequential anionic polymerization; the second strategy was based on arn“Click-Chemistry” coupling reaction. In a following step glycidol was hypergraftedrnfrom these block copolymers by applying a hypergrafting reaction with glycidol. Thernbehavior of the amphiphilic block copolymers synthesized was studied in differentrnsolvents. Furthermore the polarity of the solvent was changed to form the correspondingrninverse micelles. DLS, SLS, SEC-MALLS-VISCO, AFM and Cyro TEMrnmeasurements were performed to obtain a visual image from the appearance of thernaggregates. It was found that a linear-hyperbranched architecture is necessary, ifrnwell defined, monodisperse aggregates are required, e.g. for the preparation of orderedrnnanoarrays. Linear-linear block copolymers formed only polydisperse aggregates.rnAdditionally it was found that size distribution could be improved dramaticallyrnby passing the aggregates through a SEC column with large pores. The SEC columnsrnacted like a template in which the aggregates adopt a more stable conformation.rnIn the second part anionic polymerization was employed to synthesize silaneendfunctionalizedrnmacromonomers with different molecular weights based on polybutadienernand polyisoprene. These were polymerized by a hydrosilylation reaction inrnbulk to obtain branched polymers, using Karstedt’s catalyst. Surprisingly the additionrnof monofunctional silanes during the polymerization had only a minimal effect concerningrnthe degree of polymerization. It was possible to introduce silanes without increasingrnthe overall number of reaction steps by a very convenient “pseudo-copolymerization”rnmethod. All branched polymers were analyzed by SEC, SEC-MALLS,rnSEC-viscometry, 1H-NMR-spectroscopy and DSC concerning their branching ratio.rnThe branching parameters for the branched polymers exhibited similar characteristicsrnas hyperbranched polymers based on AB2 monomers. Detailed kinetic study showedrnthat the polymerization occurred very rapidly in comparison to the hydrosilylation polymerizationrnof classical AB2 type carbosilanes monomers.rnThe last part will deal with ferrocenyl-functionalized polymers. On the one hand,rnferrocenyl-functionalized polyglycerols (PG) were studied. Esterification of PGs withrndifferent molecular weight using ferrocenemonocarboxylic acid gave the ferrocenylrnfuntionalized polymers in high yields. On the other hand three different block copolymersrnwere prepared with different ratios of styrene to butadiene units (10:1, 4:1, 2:1).rnThe double bonds of the 1,2-PB block were hydrosilylated using silanes bearing onern(HSiMe2Fc) or two (HSiMeFc2) ferrocene units. High degrees of functionalizationrnwere obtained (up to 83 %). In this manner, six different ferrocenyl-rich block copolymersrnwith different fractions of ferrocene were prepared and analyzed, employingrnNMR-spectroscopy, SEC, SEC/MALLS/viscometry, DLS and cyclic voltammetry. Thernredox properties of the studied polymers varied primarily with the nature of the silanernunit attached. Additionally, the redox properties in solution of the studied polymersrnwere influenced by the block length ratio of the block copolymers. Unexpectedly, withrnincreasing block length of the ferrocenyl block the fraction of active ferrocenes decreased.rnNevertheless, in case of thin monolayer films this behaviour was not observed.rnAll polymers (PG and PS-b-PB based) exhibited good electrochemical propertiesrnin a wide range of solvents, which rendered them very interesting for biosensoricrnapplications.

Ergodicity and regularity of invariant measure for branching Markov processes with immigration

In this thesis we consider systems of finitely many particles moving on paths given by a strong Markov process and undergoing branching and reproduction at random times. The branching rate of a particle, its number of offspring and their spatial distribution are allowed to depend on the particle's position and possibly on the configuration of coexisting particles. In addition there is immigration of new particles, with the rate of immigration and the distribution of immigrants possibly depending on the configuration of pre-existing particles as well. In the first two chapters of this work, we concentrate on the case that the joint motion of particles is governed by a diffusion with interacting components. The resulting process of particle configurations was studied by E. Löcherbach (2002, 2004) and is known as a branching diffusion with immigration (BDI). Chapter 1 contains a detailed introduction of the basic model assumptions, in particular an assumption of ergodicity which guarantees that the BDI process is positive Harris recurrent with finite invariant measure on the configuration space. This object and a closely related quantity, namely the invariant occupation measure on the single-particle space, are investigated in Chapter 2 where we study the problem of the existence of Lebesgue-densities with nice regularity properties. For example, it turns out that the existence of a continuous density for the invariant measure depends on the mechanism by which newborn particles are distributed in space, namely whether branching particles reproduce at their death position or their offspring are distributed according to an absolutely continuous transition kernel. In Chapter 3, we assume that the quantities defining the model depend only on the spatial position but not on the configuration of coexisting particles. In this framework (which was considered by Höpfner and Löcherbach (2005) in the special case that branching particles reproduce at their death position), the particle motions are independent, and we can allow for more general Markov processes instead of diffusions. The resulting configuration process is a branching Markov process in the sense introduced by Ikeda, Nagasawa and Watanabe (1968), complemented by an immigration mechanism. Generalizing results obtained by Höpfner and Löcherbach (2005), we give sufficient conditions for ergodicity in the sense of positive recurrence of the configuration process and finiteness of the invariant occupation measure in the case of general particle motions and offspring distributions.

Non-parametric estimation of the diffusion coefficient of a branching diffusion with immigration

Wir betrachten Systeme von endlich vielen Partikeln, wobei die Partikel sich unabhängig voneinander gemäß eindimensionaler Diffusionen [dX_t = b(X_t),dt + sigma(X_t),dW_t] bewegen. Die Partikel sterben mit positionsabhängigen Raten und hinterlassen eine zufällige Anzahl an Nachkommen, die sich gemäß eines Übergangskerns im Raum verteilen. Zudem immigrieren neue Partikel mit einer konstanten Rate. Ein Prozess mit diesen Eigenschaften wird Verzweigungsprozess mit Immigration genannt. Beobachten wir einen solchen Prozess zu diskreten Zeitpunkten, so ist zunächst nicht offensichtlich, welche diskret beobachteten Punkte zu welchem Pfad gehören. Daher entwickeln wir einen Algorithmus, um den zugrundeliegenden Pfad zu rekonstruieren. Mit Hilfe dieses Algorithmus konstruieren wir einen nichtparametrischen Schätzer für den quadrierten Diffusionskoeffizienten $sigma^2(cdot),$ wobei die Konstruktion im Wesentlichen auf dem Auffüllen eines klassischen Regressionsschemas beruht. Wir beweisen Konsistenz und einen zentralen Grenzwertsatz.