Hierarchical nanophase-separated structures created by precisely-designed polymers with complexity

AbstractThis review paper summarizes recent advances in self-assembly of complex polymers, focusing on three characteristic polymeric systems. The first is star-branched polymers of the ABC type, the second one consists of multiblock terpolymers with different chain lengths, while the third comprises supramacromolecular assembly systems with hydrogen and ionic bonding interactions between different polymer species. A quasicrystalline tiling structure with mesoscopic length scale has been found for the first star polymer system as well as the periodic Archimedean tiling structures, and moreover three-dimensional Zincblende network structure has been discovered. Furthermore the hierarchical structures having two length scales have been also found for the ABC star molecules whose chain length ratios, that is, A/B and/or A/C are larger than approximately five. Hierarchical structures with double periodicity have been observed for the hexablock and undecablock terpolymers and it has been revealed that their morphology changes systematically depending on composition of polymeric species. Poly(4-hydroxystyrene) (H) homopolymer was found to be dissolved into microdomain of poly(2-vinylpyridine) formed by poly(styrene-b-2-vinylpyridine) due to hydrogen bonding interaction, resulting in the origin of morphological transitions depending on the composition of H homopolymer added. Hierarchical structures possessing double periodicity have been found for poly(isoprene-b-2-vinylpyridine)/poly(styrene-b-4-hydroxystyrene) blends depending on both volume fractions of component polymers and blend ratio. Blends of different homopolymers with several complementary nucleotides or acid/base moieties on chain ends have been confirmed to show nanophase-separated structures as a result of successful formation of “supramacromolecules”

Analytical solutions describing the phase separation driven by a free energy functional containing a long-range interaction term

We are primarily concerned with the variational problem with long-range interaction. This functional represents the Gibbs free energy of the microphase separation of diblock copolymer melts. The critical points of this variational problem can be regarded as the thermodynamic equilibrium state of the phase separation phenomenon. Experimentally it is well-known in the diblock copolymer problem that the final equilibrium state prefers periodic structures such as lamellar, column, spherical, double-diamond geometries and so on. We are interested in the characterization of the periodic structure of the global minimizer of the functional (corresponding to the strong segregation limit). In this paper we completely determine the principal part of the asymptotic expansion of the period with respect to ε (interfacial thickness), namely, we estimate the higher order error term of the period with respect to ε in a mathematically rigorous way in one space dimension. Moreover, we decide clearly the dependency of the constant of proportion upon the ratio of the length of two homopolymers and upon the quench depth. In the last section, we study the time evolution of the system. We first study the linear stability of spatially homogeneous steady state and derive the most unstable wavelength, if it is unstable. This is related to spinodal decomposition. Then, we numerically investigate the time evolution equation (the gradient flow of the free energy), and see that the free energy has many local minimizers and the system have some kind of sensitivity about initial data