Shape of Domains in Two-Dimensional Systems: Virtual Singularities and a Generalized Wulff Construction

We report on a generalized Wulff construction that allows for the calculation of the shape of two-dimensional materials with orientational order but no positional order. We demonstrate that for sufficiently large domain radii, the shape necessarily develops mathematical singularities, similar to those recently observed in Langmuir monolayers. The physical origin of the cusps is shown to be related to the softness of the material and is fundamentally different from that of the sharp angles seen in the shape of hard crystals.