Abstract

Topological transformations such as neighbour switches or bubble extinction play an important role in foams. They contribute to the onset of disorder during coarsening or flow; they provide fast elementary mechanisms for dissipation; they explain, to a large extent, the particular visco-plastic response of foams and similar mesophases. Geometrically, topological moves provide elementary steps to pass from one structure to another, and so offer the possibility to design new models from know templates.

The 3-arm star shaped molecules—sometimes called mikto arm polymers or linactant—with mutually non-miscible branches self-assemble in a variety of morphologies, giving rise to a rich phase diagram. The models for these mesophases can be interpreted as partitions of space into three coloured domains satisfying particular rules. The molecular cores aggregate along triple lines at the junction of the three coloured domains.

The topological transformations occur when two or more triple line segments come into contact, generating an unstable vertex. We will see the physical reason for this instability. We will also analyse the various ways into which a vertex can relax. The simplest cases can be explored systematically, but the complexity increases rapidly. We will consider the ingredients of a possible classification. The vertex type is characterised by a sphere pattern, induced by the 3-coloured partition on a small sphere surrounding the vertex. The class of sphere patterns can be spanned by operating 2D elementary topological transformations compatible with the colouring.