Searching for quasicrystals in block copolymer phase separation

Quasicrystals are materials that display long-range order despite lacking translational periodicity. Despite it has been 41 years since its discovery, the stability of quasicrystals remains a perplexing enigma for scientists. Initially discovered in metals, these structures also appear in soft matter systems like block copolymers. Polymer chains that contain two or more different types of monomer blocks joined together are block copolymers. They can microphase separate to form different patterns and structures in their morphology, including quasicrystals. Different morphological structures are formed depending on the block lengths and the interaction strengths. This soft matter system, akin to a designer material, can be adjusted to autonomously self-assemble into various intriguing morphologies, including quasicrystals. This thesis proposes two methods for designing block copolymers with the potential to self-assemble into quasicrystals. The stability of morphologies in block copolymers can be determined using well-established phase separation theories: weak segregation theory (WST), self-consistent field theory (SCFT), and strong segregation theory (SST). In this study, we present design criteria for two categories of block copolymers: two-component alternating linear chains and ABC star terpolymers within the context of weak segregation limit. These criteria guide the self-assembly of structures with length-scale ratios conducive to quasicrystals. The second half of the thesis presents a novel framework in strong segregation limit where morphologies in $ABC$ star terpolymers are compared with tiling patterns to study their stability. This framework can incorporate periodic tilings and periodic approximants of aperiodic tilings and develop the phase space for ABC star terpolymers. The overarching aim is to make experimentally valid predictions on polymer architectures that could lead to stable 2- and 3-dimensional quasicrystals or other structures. Using the two methodologies, we find experimentally feasible composition ranges in the block copolymers we are considering in this thesis that can potentially form quasicrystal or other interesting, complex morphologies.