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 diffferent from that of the sharp angles seen in the shape of hard crystals. 61.30.Cz, 68.10.-m, 68.35.Md, 82.65.Dp Typeset using REVTEX 1 The singularities in the shape of crystalline materials have long been understood to be a macroscopic expression of the positional order of crystals at the atomic level. The angles between the faces of a crystal are, for instance, structural invariants dependent only on a certain set of integers (Miller indices [1]). In a classic paper [2], Wulff developed a geometrical construction that allows one to determine the equilibrium shape of a crystal, provided one knows the values of the surface energies for the various Miller indices. If, for every index, the corresponding surface is placed a distance from a fixed point proportional to the surface energy, then the inner envelope of the planes constitutes the minimum energy crystal shape. Surprisingly, sharp edges in the shapes of samples are not just encountered for hard, crystalline materials. Using the Wulff construction, Herring argued that liquid crystals without positional order, but with orientational order could also display sharp ables, or cusps [3]. The cusps are no longer material invariants. Cusped domain shapes are, indeed, welldocumented for three dimensional liquid crystals [4,5]. More recently, studies of the shapes of two dimensional materials without positional order have also revealed cusps and sharp angles [6]. These studies have also revealed that the internal structure of such “soft” materials is significantly deformed and dependent on the domain shape, while Herring assumed a rigid internal structure. Thus, the Wulff construction is not manifestly valid in this case. In this paper we present a formalism which allows for the construction of domain shapes of soft two dimensional materials when the internal strucure is described by a two dimensional XY model (e. g. hexatic or nematic liquid crystals). We will demonstrate that cusps are, indeed, not just possible, but that they ought to be a generic feature of such domain shapes. Moreover, the cusp angle provides important information on the elastic moduli and surface energy of the material. We will model the internal structure of a domain by a unit vector ĉ = (cosΘ(~r), sinΘ(~r)). The associated XY model free energy is H [Θ(x, y)] = ∫ κ 2 ∣