Square-triangle tilings: an infinite playground for soft matter.

Regular square and triangle, two very simple geometrical figures, can be used to construct a fascinating variety of tilings which cover the 2D plane without any overlaps or holes. Such tilings are observed in many soft matter systems. Here we present a way to describe all possible globally uniform square-triangle phases using a three dimensional composition space. This approach takes into account both the overall composition and the orientations of the two kinds of tiles. The geometrical properties of special phases encountered in soft matter systems are described: the Archimedean Σ and H phases, the striped phases and the 12-fold maximally symmetric phases. We show how this very rich behavior with either periodic or aperiodic phases appears here as a consequence of the inherent incommensurability between the areas of the two tiles related by the ratio . Geometrical constraints on boundary lines and junction points between domains of different compositions are given, a situation likely to be encountered in experimental and numerical studies. Future developments are suggested like considering the effect on phase behavior of possible symmetry breaking.

Infinite-pressure phase diagram of binary mixtures of (non)additive hard disks.

One versatile route to the creation of two-dimensional crystal structures on the nanometer to micrometer scale is the self-assembly of colloidal particles at an interface. Here, we explore the crystal phases that can be expected from the self-assembly of mixtures of spherical particles of two different sizes, which we map to (additive or non-additive) hard-disk mixtures. We map out the infinite-pressure phase diagram for these mixtures using Floppy Box Monte Carlo simulations to systematically sample candidate crystal structures with up to 12 disks in the unit cell. As a function of the size ratio and the number ratio of the two species of particles, we find a rich variety of periodic crystal structures. Additionally, we identify random tiling regions to predict random tiling quasicrystal stability ranges. Increasing non-additivity both gives rise to additional crystal phases and broadens the stability regime for crystal structures involving a large number of large-small contacts, including random tilings. Our results provide useful guidelines for controlling the self-assembly of colloidal particles at interfaces.

Non-Fermi-Liquid Behavior in Metallic Quasicrystals with Local Magnetic Moments.

Motivated by the intrinsic non-Fermi-liquid behavior observed in the heavy-fermion quasicrystal Au51Al34Yb15, we study the low-temperature behavior of dilute magnetic impurities placed in metallic quasicrystals. We find that a large fraction of the magnetic moments are not quenched down to very low temperatures T, leading to a power-law distribution of Kondo temperatures P(T(K))∼T(K)(α-1), with a nonuniversal exponent α, in a remarkable similarity to the Kondo-disorder scenario found in disordered heavy-fermion metals. For α<1, the resulting singular P(T(K)) induces non-Fermi-liquid behavior with diverging thermodynamic responses as T→0.

Quantum antiferromagnetism in quasicrystals.

The antiferromagnetic Heisenberg model is studied on a two-dimensional bipartite quasiperiodic lattice. Using the stochastic series expansion quantum Monte Carlo method, the distribution of local staggered magnetic moments is determined on finite square approximants with up to 1393 sites, and a nontrivial inhomogeneous ground state is found. A hierarchical structure in the values of the moments is observed which arises from the self-similarity of the quasiperiodic lattice. The computed spin structure factor shows antiferromagnetic modulations that can be measured in neutron scattering and nuclear magnetic resonance experiments. This generic model is a first step towards understanding magnetic quasicrystals such as the recently discovered Zn-Mg-Ho icosahedral structure.