Coil-ring-coil block copolymers as building blocks of supramolecular hollow cylindrical brushes

This work describes the synthesis of a new class of rod-coil block copolymers, oligosubstituted shape persistent macrocycles, (coil-ring-coil block copolymers), and their behavior in solution and in the solid state.The coil-ring-coil block copolymers are formed by nanometer sized shape persistent macrocycles based on the phenyl-ethynyl backbone as rigid block and oligomers of polystyrene or polydimethylsiloxane as flexible blocks. The strategy that has been followed is to synthesize the macrocycles with an alcoholic functionality and the polymer carboxylic acids independently, and then bind them together by esterification. The ester bond is stable and relatively easy to form.The synthesis of the shape persistent macrocycles is based on two separate steps. In the first step the building blocks of the macrocycles are connected by Hagiara-Sogonaschira coupling to form an 'half-ring' as precursor, that contains two free acetylenes. In the second step the half-ring is cyclized by forming two sp-sp bonds via a copper-catalyzed Glaser coupling under pseudo-high-dilution conditions. The polystyrene carboxylic acid was prepared directly by siphoning the living anionic polymer chain into a THF solution, saturated with CO2, while the polydimethylsiloxane carboxylic acid was obtained by hydrosilylating an unsaturated benzylester with an Si-H terminated polydimethylsiloxane, and cleavage of the ester. The carbodiimide coupling was found to be the best way to connect macrocycles and polymers in high yield and high purity.The polystyrene-ring-polystyrene block copolymers are, depending on the molecular weight of the polystyrene, lyotropic liquid crystals in cyclohexane. The aggregation behavior of the copolymers in solution was investigated in more detail using several technique. As a result it can be concluded that the polystyrene-ring-polystyrene block copolymers can aggregate into hollow cylinder-like objects with an average length of 700 nm by a combination of shape complementary and demixing of rigid and flexible polymer parts. The resulting structure can be described as supramolecular hollow cylindrical brush.If the lyotropic solution of the polystyrene-ring-polystyrene block copolymers are dried, they remain birefringent indicating that the solid state has an ordered structure. The polydimethylsiloxane-ring-polydimethylsiloxane block copolymers are more or less fluid at room temperature, and are all birefringent (termotropic liquid crystals) as well. This is a prove that the copolymers are ordered in the fluid state. By a careful investigation using electron diffraction and wide-angle X-ray scattering, it has been possible to derive a model for the 3D-order of the copolymers. The data indicate a lamella structure for both type of copolymers. The macrocycles are arranged in a layer of columns. These crystalline layers are separated by amorphous layers which contain the polymers substituents.

The geometry of Lagrangian fibres

If the generic fibre f−1(c) of a Lagrangian fibration f : X → B on a complex Poisson– variety X is smooth, compact, and connected, it is isomorphic to the compactification of a complex abelian Lie–group. For affine Lagrangian fibres it is not clear what the structure of the fibre is. Adler and van Moerbeke developed a strategy to prove that the generic fibre of a Lagrangian fibration is isomorphic to the affine part of an abelian variety.rnWe extend their strategy to verify that the generic fibre of a given Lagrangian fibration is the affine part of a (C∗)r–extension of an abelian variety. This strategy turned out to be successful for all examples we studied. Additionally we studied examples of Lagrangian fibrations that have the affine part of a ramified cyclic cover of an abelian variety as generic fibre. We obtained an embedding in a Lagrangian fibration that has the affine part of a C∗–extension of an abelian variety as generic fibre. This embedding is not an embedding in the category of Lagrangian fibrations. The C∗–quotient of the new Lagrangian fibration defines in a natural way a deformation of the cyclic quotient of the original Lagrangian fibration.