Shape of domains in twodimensional systems:
virtual singularities
and
a generalized Wulff construction
Abstract
We report on a generalized Wulff construction that allows for the calculation of the shape of twodimensional 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.
pacs:
61.30.Cz, 68.10.m, 68.35.Md, 82.65.DpThe 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 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 . The associated model free energy is
(1) 
Here, is the stiffness (or Frank constant) of the order parameter field and is the anisotropic surface energy, which depends on the relative angle, , between the order parameter and the outward normal to the domain boundary. Note that .
To find the optimal domain shape we must minimize with respect to and the domain shape, while keeping the domain area, , fixed. We start with the limiting case when the texture is rigid so that and only the shape needs to be varied.
The domain shape is a closed planar curve. We will express the coordinates of a point on the curve in terms of the angle of and the minimum distance between the tangent line through and the origin. In terms of :
(2a)  
(2b) 
The variational equation determining the domain shape is, then
(3) 
The first term above is just the surface energy, with a line element along the domain boundary. The second terms is , with a Lagrange multiplier. The variational equation reduces to
(4) 
with as the solution, modulo an overall translation of the domain. The domain shape corresponds precisely to the Wulff construction. The somewhat unusual parameterization of two dimensional curves thus allows for a straightfoward and analytical determination of the domain shape.
The results above have been derived previously [7]. However, it has not, to our knowledge, been noted that variational derivatives with respect to can be carried out when the anisotropic surface energy of an element of boundary depends on the location of the element as well as its orientation. One simply replaces by and parameterizes and as in Eqs. (2). In the remainder of this Letter we describe the consequences of this strategy as applied to a twodimensional domain.
finite
We now allow the internal structure of the domain to respond to changes in the domain shape. Minimizing with respect to the angle leads to the requirement
(5a)  
(5b) 
In Eq. (5b) the derivative is along the outward normal. With no loss of generality we can write the solution of Eq. (5a) as
(6) 
with an arbitrary analytic function. It is convenient to rescale and by the mean domain radius . The domain boundary in the complex plane of a nearly circular domain is, then, the unit circle . Eq. (5b) then reduces to
(7) 
with on the unit circle.
If we now go through the same free energy minimization as for , we find instead of Eq. (4)
(8) 
with an isotropic surface tension and
(9)  
Equations (8) and (9), the latter relation holding if the domain is not too deformed from a perfect circle, are our key results. If we know the function on the unit circle—through Eq. (7)—then we can reconstruct the domain shape with the use of Eqs. (8) and (9). Note from the discussion following Eq. (4) that we can interpret the solution of Eq. (8) as being proportional to the effective surface tension, i. e. a surface tension that incorporates the interior softness of the domain.
Finding the complex function looks like an intractable problem, as Eq. (7) is highly nonlinear. We will consider some physically relevant forms for to show how and the domain shape may be found.
The case corresponds to an anisotropic surface energy proportional to . The case corresponds to the anisotropy energy of a two dimensional nematic.
It can be verified, by direct substitution, that
(10) 
is a solution of Eq. (7), with
(11) 
When the texture corresponds to the twodimensional version of a “virtual boojum”[8, 9]—a singularity in the texture that lies outside of the domain. Imaging of domains of monolayer and nearmonolayer films by Brewster angle microscopy reveals strong evidence for the existence of this structure. [10, 11, 12]. The distance of the boojum from the center of the domain, , is given by
(12) 
If the parameter is small compared to one, then approaches , while in the limit the boojum retreats to infinity. The texture is, clearly, highly deformed when .
The domain shape is found by substituting from Eq. (10) into Eq. (9). Surprisingly, , so . The domain shape is thus a perfect circle for . For the exact solution for the texture is equivalent to the texture produced by singularities lying outside of the domain. These singularities are no longer boojums in the accepted sense of the term [8, 9]. The order parameter angle advances by as one traces a path encircling one of the singularities; an advance of characterizes the boojum. The singularities for are “fractionally charged” in the topological sense. However, the fractional nature of these singularities appears to have no consequence for the textural structure of interest, as the singularities lie outside of the domain. Now is no longer equal to zero. The domain shape is deformed from perfect circularity.
This is the anisotropic boundary energy believed to be relevant to domains of liquidcondensed phase in Langmuir monolayers. It represents the lowest three terms in a systematic Fourier expansion of the anisotropic surface energy. For , the function is given by
(13) 
as can be checked by direct substitution. The small admixture of in has dramatic effects on the shape. Using Eq. (13) in Eqs. (8) and (9), we find that

For , the domain has a smooth, nearly circular, shape.

For , the domain has a cusp singularity in its shape. For , the cusp angle —the difference between the inner angle of the cusp and —obeys
(14) Note that the asymptotic variation in the excluded angle is independent of the parameter . The term is thus a singular perturbation. The domain remains nearly circular outside of the immediate vicinity of the cusp.
Large domains will, thus, always have a cusped boundary. Figure 1 graphs the evolution of the excluded angle of the cusp as a function of the radius of a domain in which surface and bulk energy parameters have values that are consistent with the expansion described above. Figure 2 depicts a cusped domain as generated by the generalized Wulff construction for a specific domain radius. The inner angle of the cusp, , is indicated in the Figure.
The measurement of cusp angle can be achieved by visual inspection of the domain. In light of the results reported above, a plot of versus should yield the parameter ratio , and also the combination . A measurement of the cusp angle as a function of domain radius has, in fact, been carried out[13]. The results will be reported elsewhere.
In summary, we have found that deformable domains of materials with an like order parameter may have shape singularities, even though their texture is perfectly analytic in the domain of definition. It is important to note that the material softness plays a key role. If we set the stiffness, , to infinity, then the boundary does not have a cusp. The physical origin of our cusp is thus strikingly different from that of hard crystalline materials. It would be interesting to know whether the results reported here extend to three dimensions.
Acknowledgement is due to Dr. Daniel Schwartz and Professor Charles Knobler for very useful discussions. J. R. would like to acknowledge the hospitality of the Institute for Theoretical Physics at the Chalmers Institute of Technology and the Service de Physique Théorique at Saclay, where some of this work was carried out.
References
 [1] See, for example, C. Kittel Introduction to Solid State Physics, 6th ed. (John Wiley and Sons, New York, 1886).
 [2] G. Wulff, Z. Krist 34, 449 (1901).
 [3] C. Herring in Structure and Properties of Solid Surfaces, R. Gomer and C. Smith, eds. (University of Chicago Press, Chicago, 1953).
 [4] G. Friedel and F. Grandjean, Bull Soc. Fr. Mineral 33, 409 (1910); J. B. Fournier and G. Durand, J. Phys. II Fr. 1, 845 (1991).
 [5] P. Oswald and F. Melo, J. de Physique 50, 3527 (1989).
 [6] H. Bercegol, F. Gallet, D. Langevin and J. Meunier, J. Phys. Fr. 50, 2277 (1989).
 [7] W. K. Burton, N. Cabrera and F. C. Frank, Philos. Trans. Roy. Soc. 243, 299 (1951).
 [8] S. A. Langer and J. P. Sethna, Phys. Rev. A 34 5035 (1986). For the original discussion of boojums, see ref. [9].
 [9] N. D. Mermin in Quantum Fluids and Solids, S. B. Trickey, E. Adams and J. Duffy, eds. (Plenum, New York, 1977).
 [10] D. Hönig, G. A. Overbeck, and D. Möbius, Adv. Mater. 4, 419 (1992).
 [11] J. Meunier, unpublished.
 [12] See also the polarized reflection micrograph of a freely suspended liquid crystal film by N. A. Clark, D. H. Van Winkle and C. Muzny in ref. [8].
 [13] D. Schwartz and C. Knobler, unpublished.