Pentagonal tiling

Abstract: We present a systematic scanning tunneling microscopy study on the 2D self-assembly of pentagonal shaped macrocycles at the liquid/solid interface to gain insight about 2D pentagonal molecular crystallization. Macrocycles with alkyl ester groups form an anti-parallel type linear pattern mainly via favorable intermolecular interactions between the interdigitated alkyl chains at the 1,2,4-trichlorobenzene (TCB)/graphite interface. On the other hand, at the TCB/Au(111) interface, they showed a random pattern and an anti-parallel type linear pattern. The selection between both motifs is ascribed to the balance between the intermolecular and molecule–substrate interactions. Most interestingly, the parent molecule forms dense anti-parallel and parallel linear patterns reflecting the intrinsic pentagonal core shape. Moreover, bright and mobile spots were often observed in the networks to occupy free spaces, including the cavity of the macrocycles, which we attribute to a co-adsorbed solvent molecule or Au ion.

Abstract: We present scanning tunnelling microscopy studies and first principles calculation on the 2D crystallization of pentagonal pentamethylcorannulene on a Cu(111) surface under ultrahigh vacuum in the temperature range of 50 K to 400 K. The observed 2D crystal phases and their packing densities are compared to tiling options of hard pentagons. Temperature change-induced reversible phase transitions reveal entropic effects in 2D crystallization. Only inclusion of dispersion interactions into density functional theory yields structures observed experimentally at low temperatures.

Abstract: We analyze substitution tiling spaces with fivefold symmetry. In the substitution process, the introduction of randomness can be done by means of two methods which may be combined: composition of inflation rules for a given prototile set and tile rearrangements. The configurational entropy of the random substitution process is computed in the case of prototile subdivision followed by tile rearrangement. When aperiodic tilings are studied from the point of view of dynamical systems, rather than treating a single one, a collection of them is considered. Tiling spaces are defined for deterministic substitutions, which can be seen as the set of tilings that locally look like translates of a given tiling. Cech cohomology groups are the simplest topological invariants of such spaces. The cohomologies of two deterministic pentagonal tiling spaces are studied.