Explaining complex metals with polymers.

Analogies and metaphors are widespread in science. We use them to convey difficult concepts to colleagues or when popularizing science to communicate with the general public. Analogies also form an integral part of the scientific process itself (1). 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. Lee et al. (2) study a system of nearly identical diblock copolymers. 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. When an initially disordered liquid state, made up of these fuzzy little balls, is cooled down, one observes a phase transition into an ordered state with a simple body-centered cubic (bcc) structure. Upon further cooling, the system undergoes a secondary phase transition into a more complex structure with tetragonal symmetry (that of a square prism), called a Frank–Kasper σ phase (Fig. 1). It is the latter transition from a cubic phase to a lower-symmetry tetragonal phase—which is presumed to be the ground-state configuration that persists down to the lowest temperatures—that is the focus of Lee et al.’s (2) analogy with metals. The bcc phase consists of cubic cells with one micelle at each corner and an additional micelle at the center of the cube. Because each corner-micelle is shared by eight cubic cells, one has a total of only two micelles per unit cell, with each micelle seeing the same environment. 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. Because of the energetic cost for having spatial variations in density, the micelles in the low-symmetry phase end up having five distinct sizes, ranging from 176 to 206 chains depending on their volumes. 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? Years ago, Frank and Kasper (4, 5) were attempting to explain the complex structures that appear in metals, using a rather simple geometric requirement for the efficient packing of hard spheres in three dimensions—yet another analogy. Arguing that because the closest arrangement of three hard spheres is a triangle, and that of four spheres is a tetrahedron, Frank and Kasper arrived at four “normal coordination polyhedra” that would typically surround a single sphere. General principles, regarding the different ways one could combine these frequently occurring Frank– Kasper polyhedra into space-filling structures, led Frank and Kasper to discover a number of typical phases, one of which is the σ phase. Frank and Kasper’s characterization of metal atoms as hard spheres is extremely successful in explaining many of the structures one commonly observes in metallic alloys. Could there be an equally useful analogy, to soft rather than hard spheres, providing additional insight into the same problem? 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