Three-colour pride—geometry and topological transformations in self-assembly models of three-arm star molecules

Abstract Polyphilic three-arm star-shaped molecules self-assemble in a wide variety of morphologies. We present an analysis of the geometry of the possible mesophases formed by three-star molecules with mutually immiscible arms. The structures resulting from such molecular self-assembly can be interpreted as partitions of space into three-coloured domains obeying geometrical principles that will be summarized here. The design of morphological templates often involves structural adjustments. This paper investigates elementary topological transformations, how they contribute to relaxing unstable local configurations and the physical reason for this instability. In particular, a vertex can relax in various ways that are explored systematically. It is conjectured that similar dissipative morphological changes may be involved in the rheology and visco-plastic response of these heterogeneous fluids. In analogy with T1 topological changes in soap froths, a cT1 move is introduced, providing elementary conservative topological transformations in three-coloured partitions. This article is part of the theme issue ‘Geometry, materials and the imagination’.