Random and ordered phases of off-lattice rhombus tiles

TL;DR

This paper investigates how off-lattice rhombus tiles self-assemble into various phases, revealing phase transitions and the influence of shape and interactions on order, with implications for molecular tiling experiments.

Contribution

It introduces a comprehensive numerical study of off-lattice rhombus tilings, exploring phase behavior beyond traditional lattice models with novel insights into self-assembly and order control.

Findings

Phase transition from fluid to solid packing at high densities

Self-assembly of broken-symmetry phases at low densities

Solid phase can be either random or crystalline depending on conditions

Abstract

We study the covering of the plane by non-overlapping rhombus tiles, a problem well-studied only in the limiting case of dimer coverings of regular lattices. We go beyond this limit by allowing tiles to take any position and orientation on the plane, to be of irregular shape, and to possess different types of attractive interactions. Using extensive numerical simulations we show that at large tile densities there is a phase transition from a fluid of rhombus tiles to a solid packing with broken rotational symmetry. We observe self-assembly of broken-symmetry phases, even at low densities, in the presence of attractive tile-tile interactions. Depending on tile shape and interactions the solid phase can be random, possessing critical orientational fluctuations, or crystalline. Our results suggest strategies for controlling tiling order in experiments involving `molecular rhombi'.