Discovery of a Frank-Kasper σ phase in sphere-forming block copolymer melts

Sphere-forming block copolymers are known to self-assemble into body-centered cubic crystals near the order-disorder transition temperature. Small-angle x-ray scattering and transmission electron microscopy experiments on diblock and tetrablock copolymer melts have revealed an equilibrium phase characterized by a large tetragonal unit cell containing 30 microphase-separated spheres. This structure, referred to as the sigma (σ) phase by Frank and Kasper more than 50 years ago, nucleates and grows from the body-centered cubic phase similar to its occurrence in metal alloys and is a crystal approximant to dodecagonal quasicrystals. Formation of the s phase in undiluted linear block copolymers (and certain branched dendrimers) appears to be mediated by macromolecular packing frustration, an entropic contribution to the interparticle interactions that control the sphere-packing geometry.