Block Copolymer Assembly: Despite their structural simplicity, block copolymers can assemble into complex and often surprising structures. Lee et al. (p. 349 ; see the Perspective by Peterca and Percec ) demonstrate the formation of a new ordered phase in a diblock copolymer and a tetrablock copolymer formed from frustrated self-assembled spherical microdomains. Remarkably, this structure contains tetragonal unit cells with 30 spheres per lattice site and is related to dodecagonal quasicrystals. This structure was first described by Frank and Kasper as a sigma phase 50 years ago and is seen in some metal alloys.

Frank–Kasper phases are tetrahedrally packed structures occurring in numerous materials, from elements to intermetallics to self-assembled soft materials. They exhibit complex manifolds of Wigner–Seitz cells with many-faceted polyhedra, forming an important bridge between the simple close-packed periodic and quasiperiodic crystals. The recent discovery of the Frank–Kasper σ-phase in diblock and tetrablock polymers stimulated the experiments reported here on a poly(isoprene-b-lactide) diblock copolymer melt. Analysis of small-angle X-ray scattering and mechanical spectroscopy exposes an undiscovered competition between the tendency to form self-assembled particles with spherical symmetry, and the necessity to fill space at uniform density within the framework imposed by the lattice. We thus deduce surprising analogies between the symmetry breaking at the body-centered cubic phase to σ-phase transition in diblock copolymers, mediated by exchange of mass, and the symmetry breaking in certain metals and alloys (such as the elements Mn and U), mediated by exchange of charge. Similar connections are made between the role of sphericity in real space for polymer systems, and the role of sphericity in reciprocal space for metallic systems such as intermetallic compounds and alloys. These findings establish new links between disparate materials classes, provide opportunities to improve the understanding of complex crystallization by building on synergies between hard and soft matter, and, perhaps most significantly, challenge the view that the symmetry breaking required to form reduced symmetry structures (possibly even quasiperiodic crystals) requires particles with multiple predetermined shapes and/or sizes. Significance Understanding how particles fill space has challenged mathematicians, scientists, and technologists since antiquity. This article rationalizes the spontaneous organization of single-component diblock copolymers into multimolecular nanoscale domains that order into a low-symmetry Frank–Kasper (FK) phase with short-range tetrahedral close packing and a giant unit cell that contains 30 particles. This class of crystal structures bridges conventional periodic crystals and low symmetry aperiodic crystals often termed “quasicrystals.” Surprising analogies are thus drawn between the heretofore unexplained formation of FK structures in soft materials, and in certain elemental metals (including manganese and uranium), alloys, and intermetallic compounds, highlighting opportunities to better understand space filling in hard and soft materials by investigation of block polymers with precisely tuned molecular architectures.

Significance: Spherical objects ranging in size from metal atoms to micron-scale colloidal particles to billiard balls tend to form regular close packed arrays with three-dimensional translational symmetry. We show that nearly spherical nanoscale micelles, formed by self-assembled diblock copolymers, spontaneously evolve into a two-dimensional dodecagonal quasicrystal following rapid cooling from the disordered liquid state. This aperiodic arrangement, characterized by rotational symmetry but not translational symmetry, transforms over time into a three-dimensional Frank–Kasper σ phase with local tetrahedral particle packing. This discovery suggests that certain forms of quasicrystalline order are nonequilibrium states generated by kinetically facilitated particle clustering in the supercooled liquid. Redistribution of polymer molecules between micelles through molecular diffusion appears to play a central role in these processes.