Rectangle-triangle soft-matter quasicrystals with hexagonal symmetry

Abstract

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.

Spontaneous pattern formation in monolayers of binary mixtures with competing interactions

“…17 Such interactions are interesting because they can lead to selfassembly into aggregates with the size determined by the shape of the potential. 2,[18][19][20][21] The models with the SALR interactions were intensively studied using theoretical and simulation methods in three (3D) and two dimensions (2D), [18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34] the latter case being suitable for the self-assembly in quasi 2D membranes at fluid interfaces or at solid substrates. The universal sequence of ordered patterns for increasing density in 2D is as follows: disordered gas -hexagonal pattern of clusters -stripes -hexagonal pattern of voids -disordered liquid.…”

“…The universal sequence of ordered patterns for increasing density in 2D is as follows: disordered gas -hexagonal pattern of clusters -stripes -hexagonal pattern of voids -disordered liquid. 18,20,25,26,35 In the gas and the liquid, randomly distributed clusters and voids with well-defined size are present. If the interactions between the membrane inclusions were of the SALR form, then clusters of well-defined size would be present.…”

Like aggregation from unlike attraction: stripes in symmetric mixtures of cross-attracting hard spheres

“…The onset of striped phases in soft matter is somewhat similar, since being related to the competition between two antagonistic forces: an isotropic short-range attraction, originated from the van der Waals and/or depletion interactions, and a long-range (but still isotropic) repulsion, arising from electrostatic forces. [13][14][15][16][17][18][19][20][21] On the other hand, it has been shown [22][23][24][25] that ''lanes'' or lamellae may even appear in onecomponent fluids interacting via purely repulsive potentials, typically modeled by a combination of hard-sphere (HS) and square-shoulder interactions. At variance with the ANNNI model, the other examples of stripes imply a genuine spontaneous symmetry breaking: the number density acquires a modulation along a definite direction which is not apparent in the system Hamiltonian.…”

Design of Linear Block Copolymers andABCStar Terpolymers That Produce Two Length Scales at Phase Separation

“…These arguments suggest that the length scales should usually be within a factor of 2 of each other to encourage QCs, with a ratio of 1.93 for 12-fold QCs and 1.618 for 10-fold or icosahedral QCs, 1,22,24,29,49,50 with other ratios stabilizing other quasicrystals. 51 Nevertheless, stable QCs can also be found with larger ratios, for example, an 8-fold quasipattern was found in a reaction−diffusion problem with a length scale ratio of 4. 26 The theoretical arguments attribute the stability of QCs to the presence of two length scales, but this presence is not sufficient for the formation of QCs: even with two length scales, hexagons, lamellae or other structures of different sizes can be stable.…”

“…For example, when the ratio of those length scales is 2 cos 15° ≈ 1.93, the nonlinear interactions between two waves of one length scale and one of the other favor density waves that are spaced 30° apart in Fourier space, − ,,, giving 12-fold symmetry. These arguments suggest that the length scales should usually be within a factor of 2 of each other to encourage QCs, with a ratio of 1.93 for 12-fold QCs and 1.618 for 10-fold or icosahedral QCs, ,,,,, with other ratios stabilizing other quasicrystals . Nevertheless, stable QCs can also be found with larger ratios, for example, an 8-fold quasipattern was found in a reaction–diffusion problem with a length scale ratio of 4 …”