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.

It is generally believed that the spherical domains self-assembled from AB-type block copolymers are composed of the minority A blocks with a volume fraction of f A < 1/2. Breaking this generic rule so that the spherical domains are formed by the majority A blocks (f A > 1/2) requires mechanisms to drastically expand the stable region of spherical packing phases. Self-consistent field theory predicts that dendron-like AB-type block copolymers, composed of G – 1 generations of A blocks connected with the outermost generation of B blocks, exhibit a stable region of spherical packing phases extending to f A ∼ 0.7. The extremely expanded spherical regions shed light on the mechanisms governing the self-assembly of amphiphilic macromolecules, as well as provide opportunities to engineer complex spherical packing phases.

The self-assembly behavior of (B1AB2)5 star copolymers, composed of five asymmetric BAB-triblock arms joined at the end of B2-blocks, has been investigated using the self-consistent field theory. The special architecture enables a few different sophisticated mechanisms such as the conformational asymmetry from the star topology, the effect of combinatorial entropy from the multiple arms that enhances the formation of bridging configurations for the core B2-blocks, the local segregation between the two different B-blocks, and the solubilization effect of the short B2-block in the majority A-domain, each of which has been individually demonstrated to play an important role in impacting the self-assembly behavior of block copolymers before. As a result, the combination of these mechanisms leads to many unusual phase behaviors of (B1AB2)5 with tunable asymmetry τ = f B1 /(f B1 + f B2 ) between the two B-blocks, where f B1 and f B2 are the volume fractions of B1- and B2-blocks, respectively. For example, reentrant phase transitions between the BCC and FCC spherical phases are observed with minority A-domains, whereas the width of the overall spherical phase region at the opposite side of the phase diagram exhibits two maxima as τ increases. The expansion of the spherical phase region at the first maximum is induced by the reduced effective volume fraction due to the solubilization effect and thus is solely occupied by the BCC phase. While the expansion at the second maximum originates from the formation of enlarged “core–shell” domains due to the effect of local segregation, leading to the formation of complex Frank–Kasper spherical phases. In addition, no stable gyroid phase composed of A-network is observed in the phase diagram of τ = 4/5, while the gyroid phase region in the opposite side of the phase diagram is expanded significantly. The absence of the gyroid phase is a very rare phenomenon for block copolymers and here may result from the combined effect of different sophisticated mechanisms.

The self-assembly of amphiphilic macromolecules into various mesocrystals has attracted abiding interest. Although many interesting mesocrystals have been achieved, mesocrystals of a low coordination number (CN) such as simple cubic are rarely reported. Here we purposely design an AB-type multiblock copolymer to target exotic spherical phases of low CNs. Self-consistent field theory reveals that two sophisticated mechanisms are realized in the copolymer, that is, stretched bridging block and released packing frustration, synergistically leading to the formation of three spherical phases with extremely low CNs, including the simple cubic spheres (CN = 6), the cubic diamond spheres (CN = 4), and normally aligned hexagonal-packing spheres (6 < CN < 8) in a considerable parameter region. Moreover, we demonstrate that these exotic phases are hard to be stabilized by either of the two mechanisms individually.

The effects of chain topology on the self-assembly of block copolymers are examined using an ABAT block copolymer, composed of an AB diblock copolymer with an extra A block tethered onto the B block, as a model system. The topology of the ABAT block copolymer is regulated by the tethering point, such that the block copolymer changes continuously from linear ABA triblock copolymer to A2B miktoarm star copolymer as the tethering position moves from the B end to the AB junction. The phase diagrams of ABAT copolymers of different tethering positions are constructed using the self-consistent field theory. The theoretical results reveal that the phase behavior of the system depends sensitively on the topology of the ABAT copolymers. In particular, a considerably wide stable region of the perforated lamellar (PL) phase is predicted for ABAT with proper tethering positions. The PL phase could even completely replaces the gyroid phase at relatively strong segregation. Furthermore, a large window of the hexagonally close-packed (hcp) spherical phase, as well as a direct transition from hcp to the cylindrical phase, is predicted. An analysis of the distributions of the different blocks reveals that the local segregation of the two different B blocks occurs to accommodate the topological constraints due to the chain architecture, which in turn regulates the local interfacial curvature and chain packing resulting in the different phase behaviors.