Combination of “living” nitroxide-mediated and photoiniferter-induced “grafting from” free-radical polymerizations: From branched copolymers to unimolecular micelles and microgels

We report on solution properties of lightly grafted copolymers composed of polystyrene (PS) backbone (degree of polymerization of PS backbone, N_b=95) and variable length of poly(tert-butyl methacrylate) P(tBuMA) side chains (degree of polymerization of side chains, N_sc=14-222) at fixed number of grafting sites n = 11 and polydispersity index (M_w/M_n) ranging from 1.05 to 2.63. Synthesis of these graft copolymers is based on a novel synthetic route [Gromadzki D, Makuška R, Netopilík M, Holler P, Lokaj J, Janata M, et al. Eur Polym J 2008;44:59-71] involving two independent controlled/“living” polymerization mechanisms, namely nitroxide-mediated radical polymerization (NMP) for the synthesis of the backbone and photoinduced “grafting from” iniferter process for building of P(tBuMA) branches. The viscosity-related contraction factors g^′<1 confirmed high degree of branching of the studied graft copolymers. Dilute solutions of graft copolymers in non-selective solvent (THF), examined by dynamic light scattering (DLS), small-angle X-ray scattering (SAXS) and viscometry, revealed a transition from comb-like conformation through wormlike-star to a microgel architecture under increasing number of monomeric units in side chains. These data were further supported by the structure factors R_η/R_h and R_g/R_h obtained by independent measurements and extrapolated to infinite dilution. Persistence lengths of the samples exhibiting comb-like topology were larger compared to linear polystyrene backbone and P(tBuMA) side chains in THF suggesting stiffening of the main chain with increasing size of the attached side chains. Unimolecular micelles were detected by DLS and SAXS in solvent selective for grafts, tert-amyl alcohol.     

Minimal genus problem for pseudo-real Riemann surfaces

A Riemann surface is said to be pseudo-real if it admits an antiholomorphic automorphism but not an antiholomorphic involution (also known as a symmetry). The importance of such surfaces comes from the fact that in the moduli space of compact Riemann surfaces of given genus, they represent the points with real moduli. Clearly, real surfaces have real moduli. However, as observed by Earle, the converse is not true. Moreover, it was shown by Seppälä that such surfaces are coverings of real surfaces. Here we prove that the latter may always be assumed to be purely imaginary. We also give a characterization of finite groups being groups of automorphisms of pseudo-real Riemann surfaces. Finally, we solve the minimal genus problem for the cyclic case.     

Morphological studies and ionic transport properties of partially sulfonated diblock copolymers

The morphological behavior of partially sulfonated polystyrene-block-poly(ethylene-alt-propylene) (PS-PEP) membranes cast from tetrahydrofuran (THF) solutions were investigated by small-angle X-ray scattering (SAXS) and atomic force microscopy (AFM).The uptakes of methanol and water increase as the sulfonation degree increases, the methanol uptake being overwhelmingly greater than the water uptake. The conductivity increases almost exponentially with increasing sulfonation degree of polystyrene units. Clusters of sulfonated units that are formed in the solution used for casting membranes persist in the solid state after evaporation. In contact with water, swelling of the membranes proceeds predominantly in these clusters. The original lamellar morphology of the diblock copolymer is progressively deformed with increasing degree of sulfonation by the presence of the clusters containing ion-rich sequences of sulfonated polystyrene blocks.     

On Ovals of Riemann Surfaces of Even Genera

Let X be a Riemann surface of genus g2. A symmetry of of X is an antiholomorphic involution acting of X. A classical theorem of Harnack states that the set Fix () of fixed points of is either emplty or it consists of g+1 disjoint simple closed curves called, following Hilberts terminology, the ovals of . A Riemann surface admitting a symmetry corresponds to a real algebraic curve and nonconjugate symmetries correspond to different real models of the curve. The number of ovals of the symmetry equals the number of connected components of the corresponding real model. It is well known that two symmetries of a Riemann surface of genus g have at most 2g+2 ovals, and the bound is attained for every genus and just for commuting symmetries. Natanzon showed that three and four nonconjugate symmetries of a Riemann surface of genus g have at most 2g+4 and 2g+8 ovals respectively, and these bounds are attained for every odd genus and for commuting symmetries. Natanzon found that a Riemann surface of genus g has at most 2( +1) nonconjugate symmetries and, again, this bound is attained for infinitely many of g. Recently we have showed that a Riemann surface of even genus g admits at most four symmetries. Our aim here is to show, using NEC groups and combinatorial methods, that three nonconjugate symmetries of a surface of even genus g has at most 2g+3 ovals and, surprisingly, if such a surface admits four nonconjugate symmetries then its total number of ovals does not exceed 2g+2. Furthermore, we show that this last bound is sharp for every even genus g and for surfaces with automorphism group D _n × Z₂, for each n dividing 2g.     

Symmetries of Accola-Maclachlan and Kulkarni surfaces

For all there is a Riemann surface of genus whose automorphism group has order , establishing a lower bound for the possible orders of automorphism groups of Riemann surfaces. Accola and Maclachlan established the existence of such surfaces; we shall call them Accola-Maclachlan surfaces. Later Kulkarni proved that for sufficiently large the Accola-Maclachlan surface was unique for and produced exactly one additional surface (the Kulkarni surface) for . In this paper we determine the symmetries of these special surfaces, computing the number of ovals and the separability of the symmetries. The results are then applied to classify the real forms of these complex algebraic curves. Explicit equations of these real forms of Accola-Maclachlan surfaces are given in all but one case.

     

Real forms of a Riemann surface of even genus

Natanzon proved that a Riemann surface of genus has at most conjugacy classes of symmetries, and this bound is attained for infinitely many genera . The aim of this note is to prove that a Riemann surface of even genus has at most four conjugacy classes of symmetries and this bound is attained for an arbitrary even as well. An equivalent formulation in terms of algebraic curves is that a complex curve of an even genus has at most four real forms which are not birationally equivalent.

     

Morphology of polystyrene-block-poly(styrene-co-acrylonitrile) and polystyrene-block-poly(styrene-co-acrylonitrile-co-5-vinyltetrazole) diblock copolymers prepared by nitroxide-mediated radical polymerization and “click” chemistry

Well-defined polystyrene-block-poly(styrene-co-acrylonitrile) PS-block-P(S-co-AN) and poly(styrene-co-acrylonitrile-co-5-vinyltetrazole) PS-block-P(S-co-AN-co-5VT) block copolymers with various content of acrylonitrile units in the statistical block were synthesized by nitroxide mediated radical polymerization (NMRP) and post-functionalized using efficient “click” chemistry process. In the second step, acrylonitrile units were successfully modified using 1,3-dipolar cycloaddition (“click” chemistry) type polymer analogue reaction. The original pristine diblock copolymers can be molecularly dissolved in THF and dioxane while the “tetrazolated” versions aggregate to clusters as determined by dynamic light scattering (DLS). Small-angle X-ray scattering (SAXS) and Transmission Electron Microscopy (TEM) revealed ordered lamellar morphology with interlamellar spacing d = 60 nm increasing to d = 80 nm for “tetrazolated” diblock copolymers. The morphological features of diblock copolymer thin layers observed by Atomic Force Microscopy (AFM) depend on the tunable content of both acrylonitrile and 5-vinyltetrazole units and on the quality (polarity) of the solvents used.     

On Real Forms of Belyi Surfaces With Symmetric Groups of Automorphisms

In virtue of the Belyi Theorem an algebraic curve can be defined over the algebraic numbers if and only if the corresponding Riemann surface can be uniformized by a subgroup of a Fuchsian triangle group. Such surfaces are known as Belyi surfaces. Here we study the actions of the symmetric groups S _n on Belyi Riemann surfaces. We show that such surfaces are symmetric and we calculate the number of connected components of the corresponding real forms.     

The full automorphism groups of hyperelliptic Riemann surfaces

For every integer g≥2 we obtain the complete list of groups acting as the full automorphisms groups on hyperelliptic Riemann surfaces of genus g. Partially supported by DGICYT PB 89-201 and Science Plan 910021 Partially supported by DGICYT PB 89/379/C02/01 and Science Plan 910021 Partially supported by DGICYT After the typing of this paper we have heard about a Ph.D. Thesis by Britta Krapp on questions related to the problem studied here.