Aperiodic approximants bridging quasicrystals and modulated structures.

Aperiodic crystals constitute a class of materials that includes incommensurate (IC) modulated structures and quasicrystals (QCs). Although these two categories share a common foundation in the concept of superspace, the relationship between them has remained enigmatic and largely unexplored. Here, we show "any metallic-mean" QCs, surpassing the confines of Penrose-like structures, and explore their connection with IC modulated structures. In contrast to periodic approximants of QCs, our work introduces the pivotal role of "aperiodic approximants", articulated through a series of k-th metallic-mean tilings serving as aperiodic approximants for the honeycomb crystal, while simultaneously redefining this tiling as a metallic-mean IC modulated structure, highlighting the intricate interplay between these crystallographic phenomena. We extend our findings to real-world applications, discovering these tiles in a terpolymer/homopolymer blend and applying our QC theory to a colloidal simulation displaying planar IC structures. In these structures, domain walls are viewed as essential components of a quasicrystal, introducing additional dimensions in superspace. Our research provides a fresh perspective on the intricate world of aperiodic crystals, shedding light on their broader implications for domain wall structures across various fields.

Programmable Self-Assembly of Nanoplates into Bicontinuous Nanostructures.

Self-assembly is the process by which individual components arrange themselves into an ordered structure by changing the shapes, components, and interactions. It has enabled us to construct an extensive range of geometric forms on many length scales. Nevertheless, the potential of two-dimensional polygonal nanoplates to self-assemble into extended three-dimensional structures with compartments and corridors has remained unexplored. In this paper, we show coarse-grained Monte Carlo simulations demonstrating self-assembly of hexagonal/triangular nanoplates via complementary interactions into faceted, sponge-like "bicontinuous polyhedra" (or infinite polyhedra) whose flat walls partition space into a pair of mutually interpenetrating labyrinths. Two bicontinuous polyhedra can be self-assembled: the regular (or Platonic) Petrie-Coxeter infinite polyhedron (denoted {6,4|4}) and the semi-regular Hart "gyrangle". The latter structure is chiral, with both left- and right-handed versions. We show that the Petrie-Coxeter assembly is constructed from two complementary populations of hexagonal nanoplates. Furthermore, we find that the 3D chiral Hart gyrangle can be assembled from identical achiral triangular nanoplates decorated with regioselective complementary interaction sites. The assembled Petrie-Coxeter and Hart polyhedra are faceted versions of two of the simplest triply periodic minimal surfaces, namely, Schwarz's primitive and Schoen's gyroid surfaces, respectively, offering alternative routes to those bicontinuous nanostructures, which are widespread in synthetic and biological materials.

Rectangle-triangle soft-matter quasicrystals with hexagonal symmetry.

Aperiodic (quasicrystalline) tilings, such as Penrose's tiling, can be built up from, e.g., kites and darts, squares and equilateral triangles, rhombi- or shield-shaped tiles, and can have a variety of different symmetries. However, almost all quasicrystals occurring in soft matter are of the dodecagonal type. Here we investigate a class of aperiodic tilings with hexagonal symmetry that are based on rectangles and two types of equilateral triangles. We show how to design soft-matter systems of particles interacting via pair potentials containing two length scales that form aperiodic stable states with two different examples of rectangle-triangle tilings. One of these is the bronze-mean tiling, while the other is a generalization. Our work points to how more general (beyond dodecagonal) quasicrystals can be designed in soft matter.

Metallic-mean quasicrystals as aperiodic approximants of periodic crystals.

Ever since the discovery of quasicrystals, periodic approximants of these aperiodic structures constitute a very useful experimental and theoretical device. Characterized by packing motifs typical for quasicrystals arranged in large unit cells, these approximants bridge the gap between periodic and aperiodic positional order. Here we propose a class of sequences of 2-D quasicrystals that consist of increasingly larger periodic domains and are marked by an ever more pronounced periodicity, thereby representing aperiodic approximants of a periodic crystal. Consisting of small and large triangles and rectangles, these tilings are based on the metallic means of multiples of 3, have a 6-fold rotational symmetry, and can be viewed as a projection of a non-cubic 4-D superspace lattice. Together with the non-metallic-mean three-tile hexagonal tilings, they provide a comprehensive theoretical framework for the complex structures seen, e.g., in some binary nanoparticles, oxide films, and intermetallic alloys.

Bronze-mean hexagonal quasicrystal.

The most striking feature of conventional quasicrystals is their non-traditional symmetry characterized by icosahedral, dodecagonal, decagonal or octagonal axes. The symmetry and the aperiodicity of these materials stem from an irrational ratio of two or more length scales controlling their structure, the best-known examples being the Penrose and the Ammann-Beenker tiling as two-dimensional models related to the golden and the silver mean, respectively. Surprisingly, no other metallic-mean tilings have been discovered so far. Here we propose a self-similar bronze-mean hexagonal pattern, which may be viewed as a projection of a higher-dimensional periodic lattice with a Koch-like snowflake projection window. We use numerical simulations to demonstrate that a disordered variant of this quasicrystal can be materialized in soft polymeric colloidal particles with a core-shell architecture. Moreover, by varying the geometry of the pattern we generate a continuous sequence of structures, which provide an alternative interpretation of quasicrystalline approximants observed in several metal-silicon alloys.

Hard spheres on the gyroid surface.

We find that 48/64 hard spheres per unit cell on the gyroid minimal surface are entropically self-organized. Striking evidence is obtained in terms of the acceptance ratio of Monte Carlo moves and order parameters. The regular tessellations of the spheres can be viewed as hyperbolic tilings on the Poincaré disc with a negative Gaussian curvature, one of which is, equivalently, the arrangement of angels and devils in Escher's Circle Limit IV.

Kaleidoscopic morphologies from ABC star-shaped terpolymers.

Star-shaped terpolymers of the ABC type composed of incompatible polymer components give a variety of ordered structures with mesoscopic length scales depending on their composition ratio. Their peculiar features are summarized in this report. Polymer components adopted are polyisoprene (I), polystyrene (S) and poly(2-vinylpyridine) (P), and many monodisperse samples of the I(X)S(Y)P(Z) type were anionically prepared. Firstly our focus is on molecules of the I(1.0)S(1.0)P(x(1)) type, where x(1) is only a variable. The complex but systematic morphology change was displayed within the range 0.2 ≤ x(1) ≤ 10, that is, their structures change from spherical plus lamellae structure for I(1.0)S(1.0)P(0.2) to periodic tilings (0.4 ≤ x(1) ≤ 1.9), then to lamellae-in-lamella (3.0 ≤ x(1) ≤ 4.9) and lamellae-in-cylinder (7.9 ≤ x(1) ≤ 10) structures with increasing x(1). Here if we pay attention to the structural variation of the P domain inclusively, it transforms from sphere to cylinder, lamella and then to matrix, which is the same as that for linear polymers. Among them, several periodic Archimedean tiling patterns can be naturally formed when the relative lengths of the three chains are close to one another. Moreover, it has been found that the tiling zone is spread out widely. For example, the series I(1.0)S(1.8)P(x(2)) (with 0.8 ≤ x(2) ≤ 2.9) and the other series I(1.0)S(y)P(2.0) (with 1.1 ≤ y ≤ 2.7) show mostly Archimedean tilings. Additionally, block copolymer/homopolymer blends with a composition of I(1.0)S(2.7)P(2.5) reveal a quasicrystalline tiling with dodecagonal symmetry. Furthermore, a zinc-blende-type four-branched network structure was created just a little outside of the tiling region for a block copolymer/homopolymer blend of I(1.0)S(2.3)P(0.8). When some more asymmetry in chain length is introduced, hyperbolic tiling on a gyroid membrane has successfully been constructed for the sample I(1.0)S(1.8)P(3.2) and it transforms into a hierarchical cylinders-in-lamella structure with further increase in P content to I(1.0)S(1.8)P(6.4). Thus, kaleidoscopic morphologies have been generated from ABC star-shaped terpolymers and their structural change has turned out to be very sensitive to relative compositions.

Polymeric quasicrystal: mesoscopic quasicrystalline tiling in ABC star polymers.

A mesoscopic tiling pattern with 12-fold symmetry has been observed in a three-component polymer system composed of polyisoprene, polystyrene, and poly(2-vinylpyridine) which forms a star-shaped terpolymer, and a polystyrene homopolymer blend. Transmission electron microscopy images reveal a nonperiodic tiling pattern covered with equilateral triangles and squares, their triangle/square number ratio of 2.3 (approximately equal to 4/sqrt[3]), and a microbeam x-ray diffraction pattern shows dodecagonal symmetry. The same kind of quasicrystalline structures have been found for metal alloys (approximately 0.5 nm), chalcogenides (approximately 2 nm), and liquid crystals (approximately 10 nm). The present result (approximately 50 nm) confirms the universal nature of dodecagonal quasicrystals over several hierarchical length scales.

Voronoi space division of a polymer: topological effects, free volume, and surface end segregation.

In order to investigate the topological effects of chain molecules, united-atom molecular dynamics simulations of a 500-mer polyethylene linked by 50 hexyl groups (a grafted polymer having 52 ends) are carried out and analyzed in terms of Voronoi space division. We find that the volume of a Voronoi polyhedron for a chain end is larger than that for an internal or junction atom, and that it is the most sensitive to temperature, both of which suggest higher mobility of chain ends. Moreover, chain ends dominantly localize at the surface of the globule: The striking evidence is that while the ratio of surface atoms is only 24% of all atoms, the ratio of ends at the surface is 91% out of all ends. The shape of Voronoi polyhedra for internal atoms is prolate even in the bulk, and near the surface it becomes more prolate. We propose the concept of bonding faces, which play a significant role in the Voronoi space division of covalently bonding polymers. Two bonding faces occupy 38% of the total surface area of a Voronoi polyhedron and determine the prolate shape.

Tricontinuous cubic structures in ABC/A/C copolymer and homopolymer blends.

Using the Monte Carlo lattice-simulation technique, we present numerical evidence of the formation of gyroid and nongyroid tricontinuous cubic phases in high polymeric systems of ABC/A/C triblock copolymer and homopolymer blends. By increasing the volume fraction of homopolymer, a remarkable phase sequence G (gyroid) --> D (diamond) --> P (primitive) is observed, which is common to certain surfactant systems. Our results indicate that the ABC triblock copolymer system with blending homopolymers may be a zoo of cubic phases, suitable for comparative studies of these phases.