Alistair Rucklidge

Soft matter quasicrystals with six-fold symmetry: pattern formation, mode interactions and tilings

Since their discovery, there has been a strong connection between quasicrystalline structures and aperiodic tilings. The inflation rule for constructing the aperiodic Penrose tiling guarantees ten-fold rotation symmetry in the power spectrum and tilings based on squares and equilateral triangles can have twelve-fold symmetry, matching the ten-fold or twelve-fold symmetry in the diffraction spectrum of metallic and soft matter quasicrystals. What is less clear is: given an aperiodic tiling, are there systems of particles that will assemble to match the tiling? Here we examine two tilings based on combinations of small and large equilateral triangles along with a rectangular tile whose edges match the two triangles. One of the two is the recently discovered Bronze-mean tiling, the other is a new variant on this. We use the diffraction spectrum of the two tilings to design particles with two length-scale soft interaction potentials, and show that the particles can assemble into structures that are local minima of their free energies and that match the two tilings.