Counting polyominoes with minimum perimeter
- Es wird die Anzahl der wesentlich verschiedenen Polyominoes der Ordnung n mit minimalem Umfang p(n) bestimmt.
Complex nanostructures in triblock terpolymer thin films
- The thin film phase behavior of poly(styrene)-block-poly(2-vinylpyridine)-block-poly(tert-butyl methacrylate) (PS-b-P2VP-b-PtBMA) triblock terpolymers with volume fractions f(PS) : f(P2VP) : f(PtBMA) = 1 : 1.2 : x, with x ranging from 3.05 to 4, is studied with a combinatorial gradient approach. Gradients in film thickness are prepared via thin film flow coating of dilute solutions in chloroform. Upon controlled annealing in nearly saturated solvent vapor the films form terraces of well-defined step height. The dependence between morphology and film thickness is studied with optical microscopy, tapping mode SFM, and SEM. Though showing different morphologies in the bulk, the same sequence of surface structures is found with increasing film thickness for the whole range of compositions: a disordered phase in the thinnest regions, a liquid-like distribution of upright standing cylinders, cylinders oriented parallel to the film, and finally a hexagonally ordered perforated lamella structure (PL) on the first terrace with a thickness of d = (37+3) nm. Higher terraces also exhibit PL as surface structures. Due to the chemical nature of the block components and the particular stoichiometry of the polymer a wetting layer with a PtBMA-rich top layer is formed next to the substrate. By imposing an additional gradient in substrate surface energy, orthogonal to the gradually increasing film thickness, the perforated lamella is shown to be a stable phase, regardless of the chemical nature of the substrate, which makes this structure and methodology robust for application in nanotechnology. The complex phase behavior observed in thin films is supported by mesoscale computer simulations based on dynamic density functional theory. Thin films of the above mentioned triblock terpolymers are modeled as a melt of A3B4C12 Gaussian chains which is confined in a slit with film thickness H. By adjusting the interaction parameters between the polymer components and the surfaces, the experimentally observed sequence of surface structures as function of the film thickness can be successfully modeled. At well-defined film thickness the perforated lamella structure is formed. In analogy with earlier work on a two-component system these structures are identified as surface reconstructions of the bulk structure. In particular, the core-shell PL can be seen as analogue to the PL surface reconstruction of cylinder-forming AB and ABA systems. The influence of film thickness, surface field, and the interaction parameters between the different polymer components on the phase behavior is also explored. A large spectrum of surface structures is observed in analogy to the experiments. Further attention has been given to the perforated lamella structure. This structure can be visualized as P2VP/PS/P2VP lamellae which are perforated by channels of PtBMA interconnecting between two outer layers of PtBMA. A highly ordered PL structure could be prepared with a very small number of defects over an area of about 12 x 4 µm2. Because of the special functionalities of the triblock terpolymer a rather versatile nanostructure was produced. By selective UV-depolymerization of the PtBMA matrix phase, the PL phase might potentially be used for lithographic applications similar to the case of perpendicularly oriented poly(methyl methacrylate) (PMMA) cylinders in PS-b-PMMA block copolymer thin films. Furthermore, a responsive membrane can be created by selective removal of the matrix phase. The remaining PL has a P2VP shell which might be either switched via the pH-value or loaded with metal components. A polymer-analogous reaction of the matrix phase of the PL to poly(methacrylic acid) via acid-catalyzed hydrolysis leads to a pH-responsive nanostructure without altering the overall structure. With SFM in aqueous environment structural changes of the PL phase are studied as function of the pH-value. Upon changing the pH of the surrounding medium a strong swelling of the original film thickness is observed at pH-values > 6 to a maximum degree of 7.5-fold swelling. This swelling is explained with a conformational change of the matrix phase poly(methacrylic acid). The hexagonal arrangement of the pattern is not affected. The first two blocks PS and P2VP act as skeleton of the PL phase which withstands the mechanical forces exerted on the strongly swollen PMAA. In contrast to the PL phase core-shell cylinders oriented parallel to the interfaces cannot withstand these forces and are solubilised at high pH-values.