Publikationen

In this paper, we exhibit connections between the following subjects: Tree packing in graphs and digraphs (both behave completely different), the rigidity matroid of a graph, Henneberg moves on trees, the conjectures of Thomassen and Matthews and Sumner, and (s,t)-orderings of digraphs. We do this by studying graphs which admit acyclic orientations that contain an out-branching and in-branching which are arc-disjoint (such an orientation is called good). A 2T-graph is a graph whose edge set can be decomposed into two edge-disjoint spanning trees. It is a well-known result due to Tutte and Nash-Williams, respectively, that every 4-edge-connected graph contains a spanning 2T-graph. Vertex-minimal 2T-graphs with at least two vertices which are known as generic circuits play an important role in rigidity theory for graphs. We prove that every generic circuit has a good orientation. Using this result we prove that if G is 2T-graph whose vertex set has a partition V1,V2, ,Vk so that each Vi induces a generic circuit Gi of G and the set of edges between different Gi's form a matching in G, then G has a good orientation. We also obtain a characterization for the case when the set of edges between different Gi's form a double tree, that is, if we contract each Gi to one vertex, and delete parallel edges we obtain a tree. All our proofs are constructive and imply polynomial algorithms for finding the desired good orderings and the pairs of arc-disjoint branchings which certify that the orderings are good. We identify a structure which can be used to certify that a given 2T-graph does not have a good orientation.

Hierarchical assembly architectures of functional polymer particles are promising because of their physicochemical and surface properties for multi-labeling and sensing to catalysis and biomedical applications. While polymer nanoparticles' interior is mainly made up of the cross-linked network, their surface can be tailored with soft, flexible, and responsive molecules and macromolecules as potential support for the controlled particulate assemblies. Molecular surfactants and polyelectrolytes as interfacial agents improve the stability of the nanoparticles whereas swellable and soft shell-like cross-linked polymeric layer at the interface can significantly enhance the uptake of guest nano-constituents during assemblies. Besides, layer-by-layer surface-functionalization holds the ability to provide a high variability in assembly architectures of different interfacial properties. Considering these aspects, various assembly architectures of polymer nanoparticles of tunable size, shapes, morphology, and tailored interfaces together with controllable interfacial interactions are constructed here. The microfluidic-mediated platform has been used for the synthesis of constituents polymer nanoparticles of various structural and interfacial properties, and their assemblies are conducted in batch or flow conditions. The assemblies presented in this progress report is divided into three main categories: cross-linked polymeric network's fusion-based self-assembly, electrostatic-driven assemblies, and assembly formed by encapsulating smaller nanoparticles into larger microparticles.

The presence of the polycationic macromolecule poly(diallyldimethylammonium chloride) (poly-DADMAC) has a strong effect on the shape and size of colloidal gold nanoparticles formed by the reduction of tetrachloroauric acid with ascorbic acid in aqueous solution. It slows down nanoparticle growth and supports the formation of nonspherical, partially highly fractal and hierarchical nanoparticle shapes. Four structural levels have been recognized from the near-spherical gold nanoparticles in the lower nanometer range over compact aggregates in the midnanometer range and flower and star-like particles in the submicron range up to larger filamentous aggregates. High-contrast scanning electron microscope (SEM) images show that single gold nanoparticles and clusters of them are connected by bundles of macromolecules in large aggregates. The investigation showed that a large spectrum of different nanoparticle shapes and sizes can be accessed by tuning the poly-DADMAC concentrations and their ratio to other reactants. The nanoassemblies with a very high specific surface area might be of interest for SERS and heterogeneous catalysis.