Influence of Thermal Fluctuation on the Stability of Frank–Kasper Spherical Phases in AB n Block Copolymers

Recently, Frank–Kasper spherical phases have attracted increasing interest in the field of block copolymers and even soft matter due to their complex packing motifs, nonuniform spherical-domain sizes, and unique thermodynamic stabilization mechanisms. Previous self-consistent field theory (SCFT) studies have shown that the stability of Frank–Kasper phases is closely related to the spontaneous curvature of block copolymers, which arises from a competition between the chain stretching of the A blocks versus the chain stretching of the B blocks. Increasing the spontaneous curvature through molecular design can broaden the stability window of spherical phases and allows spherical domains in different coordination environments to adjust their sizes to relieve packing frustration, thereby packing into the complex σ and A15 crystal lattices. However, Frank–Kasper phases were often experimentally observed in low-molecular-weight and weakly entangled block copolymers, where thermal fluctuations may play an important role. Since such fluctuations cannot be included into SCFT, their influence on the formation of Frank–Kasper phases remains unclear. Here, we employ Langevin field-theoretic simulations (L-FTS) to investigate the self-assembly of ABn (n = 2, 3, 4) miktoarm star block copolymers, focusing on how thermal fluctuations affect the stability of Frank–Kasper phases. Our results show that, at a large invariant polymerization index, such as N̅ = 105, thermal fluctuations substantially shift the order–disorder transition boundaries near the weak segregation regime but do not alter the relative stability among the σ, A15, and cylindrical phases significantly. Away from the ODT boundary, thermal fluctuations do not change the characteristics of size differences between those nonequivalent spherical domains in the Frank–Kasper phases. However, near the ODT boundary, widened interfaces together with increased thermal fluctuations may weaken the instantaneous size contrast between different domains. Overall, our results indicate that, when N̅ is large or the system is far from the ODT boundary, thermal fluctuations have a limited effect on the stability of complex Frank–Kasper spherical phases, and the SCFT predictions are reliable.