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

Abstract

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...

Explaining complex metals with polymers

“…Some may even say that science is all about finding analogies between seemingly different phenomena, organizing them using unifying abstract concepts and common models, thereby generalizing these concepts and advancing their understanding. In PNAS, Lee et al (2) draw a surprising analogy between the principles that govern the formation of ordered phases in soft-matter systems consisting of micelle-forming block copolymers (3), and those underlying the formation of hard solid-state metallic crystals.…”

“…These are molecules that consist of a pair of different polymer chains of unequal lengths attached at a single point, forming one long double-block chain. Lee et al (2) observe that under conditions that favor the segregation of the two blocks, groups of about 200 of these copolymer chains self-assemble into spherical micelles. Each micelle resembles a little ball with a core consisting mostly of one of the blocks, and a corona consisting of the other block.…”

“…In the σ phase, there are 30 micelles per square-prism cell, with a total of five distinct environments that surround the different micelles. In the bcc phase, Lee et al (2) find that all of the micelles are nearly identical, each containing almost exactly 193 diblock copolymer chains. In the σ phase, the five different environments define five volumes with different sizes and shapes, into which the micelles must fit.…”

“…Thus, a redistribution of mass between neighboring micelles must take place at the transition. Indeed, Lee et al (2) demonstrate that when cooled rapidly, so that the chains do not have sufficient time to diffuse between the micelles, the σ phase does not form. Could there be a process, which would be analogous to the exchange of mass, in similar symmetry-lowering phase transitions in single-component metals, where individual atoms are involved rather than large micelles?…”

“…Surprisingly, the answers to both of these questions may be positive. Lee et al (2) contend that the exchange of mass between micelles is analogous to an exchange of charge between atoms in metals. Specifically, the authors argue that different atoms exchange spin-dependent charges as the metal goes through the phase transition.…”