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 σ 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.

Frank-Kasper phases are tetrahedrally packed structures occurring in numerous materials, from elements to intermetallics to selfassembled 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.symmetry breaking | sphericity | Frank-Kasper phases | block polymers T he discovery of materials with aperiodic order, often referred to as "quasicrystals," 30 years ago (1, 2) heralded new and promising vistas for designing materials endowed with unique properties. In the 1950s Frank and Kasper (3, 4) recognized complex tetrahedral atomic-and molecular-packing geometries that bridge the familiar close-packed crystals [e.g., face-centered cubic (FCC), hexagonally close-packed (HCP), and body-centered cubic (BCC) structures] characterized by periodic order, and quasiperiodic crystals (QCs) that extend crystallography beyond the 230 space groups relevant to periodic crystals (5, 6). The scientific literature is replete with examples of Frank-Kasper phases in hard materials, particularly in the area of intermetallics (7-9), but also in a few complex elemental crystals, including manganese (10, 11) and uranium (12). Recently, this class of crystalline order has cropped up in a host of soft materials, including dendrimers (13), surfactant solutions (14), and block polymers (15, 16), often in close proximity to QC phases (17)(18)(19). To the best of our knowledge the principles underlying the formation of Frank-Kasper phases across both categories of materials have not been established, presenting enticing challenges to sc...

We report the discovery of a dodecagonal quasicrystalline state (DDQC) in a sphere (micelle) forming poly(isoprene-b-lactide) (IL) diblock copolymer melt, investigated as a function of time following rapid cooling from above the order-disorder transition temperature (T ODT = 66°C) using small-angle X-ray scattering (SAXS) measurements. Between T ODT and the order-order transition temperature T OOT = 42°C, an equilibrium body-centered cubic (BCC) structure forms, whereas below T OOT the Frank-Kasper σ phase is the stable morphology. At T < 40°C the supercooled disordered state evolves into a metastable DDQC that transforms with time to the σ phase. The times required to form the DDQC and σ phases are strongly temperature dependent, requiring several hours and about 2 d at 35°C and more than 10 and 200 d at 25°C, respectively. Remarkably, the DDQC forms only from the supercooled disordered state, whereas the σ phase grows directly when the BCC phase is cooled below T OOT and vice versa upon heating. A transition in the rapidly supercooled disordered material, from an ergodic liquid-like arrangement of particles to a nonergodic soft glassy-like solid, occurs below ∼40°C, coincident with the temperature associated with the formation of the DDQC. We speculate that this stiffening reflects the development of particle clusters with local tetrahedral or icosahedral symmetry that seed growth of the temporally transient DDQC state. This work highlights extraordinary opportunities to uncover the origins and stability of aperiodic order in condensed matter using model block polymers.quasicrystals | Frank-Kasper phases | block polymers T he discovery of aperiodic order, today referred to as quasicrystalline order, in rapidly cooled Al-Mn alloys by Shechtman and coworkers in 1984 heralded a paradigm shift in the understanding of what constitutes long-range order in matter (1, 2). More recently, quasicrystalline order has been found in an expanding variety of soft materials including self-assembling wedge-shaped dendrimers, mixtures of ligand coated nanoparticles, concentrated solutions of surfactant and block polymer micelles, multicomponent polymer blends containing ABC-type miktoarm star polymers, and an ABA′C-type linear multiblock polymer (3-8). The underlying principles that govern the formation of quasicrystals in soft materials remain largely unresolved.Block polymers are uniquely tunable self-assembling soft materials in which the nanoscale morphology, characteristic length scale, chemical and physical properties, and dynamics can be precisely controlled through the judicious choice of block repeat unit chemistries, block molecular weights, and molecular architecture (9, 10). Such exquisite control over structure, and the associated dynamics, makes block polymers ideal model materials for fundamental inquiries into the formation and stability of soft quasicrystals. In this article, we report the discovery of a long-lived metastable dodecagonal quasicrystalline phase (DDQC) in a (nominally) single-component poly(1,4-is...

The phase behavior for a series of poly(1,4‐isoprene‐b‐DL‐lactide) diblock copolymers characterized by a relatively large Flory–Huggins segment–segment interaction parameter (χ) and low degrees of polymerization (N) over a range of compositions was reported. Ordered‐state morphologies were deduced from small‐angle x‐ray scattering (SAXS) measurements, and χ(T) was determined from order‐disorder transition temperatures (TODT's) associated with the compositionally symmetric specimens and assuming the mean‐field theory, that is, (χN)ODT=10.5. The ODT was determined by differential scanning calorimetry, SAXS, and dynamic mechanical spectroscopy, and shown to be weakly first‐order, with latent heats of transition that vary strongly with composition. We interpret this behavior in terms of the mean and Gauss interfacial curvature of the ordered‐state morphologies and with respect to composition fluctuations in the disordered state. These results offer a fresh strategy for investigating weakly first‐order phase transitions within the Brazovskii universality class. © 2013 American Institute of Chemical Engineers AIChE J, 59: 3502–3513, 2013