The stability of various spherical phases formed in conformationally asymmetric AB diblock and architecture asymmetric AB m miktoarm block copolymers is investigated using self-consistent field theory. Both the conformational and architecture asymmetries are unified into a parameter of conformationally asymmetric degree, ε. We find that a complex spherical phase, the σ phase, becomes stable and its phase region expands between bcc and hexagonal phases as increasing ε. Only for large conformational asymmetry, for example, ε = 9 (or m = 3), the A15 phase becomes stable in the region between the σ phase and the hexagonal phase and its phase region terminates at the intermediate segregation region. Compared with the σ phase, the A15 phase has more favorable interfacial energy by enabling the formation of larger spherical domains, and therefore, it becomes more stable in the region of more symmetric volume fraction and stronger segregation.

A drop of solution containing nonvolatile solute is allowed to evaporate from a sphere-on-flat geometry. Left behind is a striking pattern of gradient concentric rings with unprecedented regularity. The center-to-center distance between adjacent rings, lambda(C-C), and the height of the ring, h(d), are strongly affected by the concentration of the solution and the properties of the solvent. The nature of the formation of regular gradient ring patterns during the course of irreversible solvent evaporation is revealed through theoretical calculations based on the mass conservation of the solution.

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The emergence of the complex Frank-Kasper phases from binary mixtures of AB diblock copolymers is studied using the self-consistent field theory. The relative stability of different ordered phases, including the Frank-Kasper σ and A15 phases containing nonspherical minority domains with different sizes, is examined by a comparison of their free energy. The resulting phase diagrams reveal that the σ phase occupies a large region in the phase space of the system. The formation mechanism of the σ phase is elucidated by the distribution of the two diblock copolymers with different lengths and compositions. In particular, the segregation of the two types of copolymers, occurring among different domains and within each domain, provides a mechanism to regulate the size and shape of the minority domains, thus enhancing the stability of the Frank-Kasper phases. These findings provide insight into understanding the formation of the Frank-Kasper phases in soft matter systems and a simple route to obtain complex ordered phases using block copolymer blends.