Sphericity and symmetry breaking in the formation of Frank–Kasper phases from one component materials

Abstract

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.