Molecular Crystals and Liquid Crystals

The problem of finding the equilibrium shape of a small particle by the Wulff construction is reviewed briefly, with emphasis on its applications to liquid crystals. The proof of Wulff’s theorem is stated in a concise mathematical form. Some typical equilibrium shapes of liquid crystalline cirops are described. When there is orientational order of the molecules in the liquid crystal but no translational order, the equilibrium shape may be an ellipsoid or a tactoid; when there is translational order as well, the shape may have plane faces, possibly with sharp edges and corners. The formation of the stepped drop, goutte a gradins, is interpreted as analogous to the stepwise roughening of a flat crystal surface whose orientation does not occur amongst the boundary surfaces of the Wulff shape. As is well known, the problem of determining the equilibrium shape of a given quantity of material reduces mathematically to one of finding the condition for minimum surface energy. It was formulated over eighty years ago by Gibbs, and independently by Curie, and has since been investigated in great detail particularly in relation to the equilibrium shapes of crystals. It is the aim of this paper to review these ideas briefly and to discuss their applications to liquid crystals. The surface tension of a liquid crystal may be expected t o be anis0tropic.l A solid crystal also has anisotropic surface tension, but there is an essential difference between the two cases.? The surface tension of a solid crystal (or the specific surface free energy, measured by the work done in creating a new surface) differs numerically from its surface stress (measured by the work done in deforming a surface). For a liquid crystal, on the other hand, these two quantities will be equal because of the inability of the medium to sustain a shear stress, except for short periods of time.3 71 D o w n l o a d e d A t : 1 2 : 2 4 2 0 J a n u a r y 2 0 1 1 72 M O L E C U L A R C R Y S T A L S Wulff’s Theorem The relation between the surface tension and the equilibrium shape of a small particle is determined by a theorem due to W ~ l f f . ~ Proofs of this theorem were given by H i l t ~ n , ~ Liebmann6 and Laue,’ all of whom assumed the equilibrium shape to be a polyhedron and compared its surface energy with that of other polyhedra with slightly different fwe areas. However, these proofs are not strictly valid for liquid crystals, for the equilibrium shape may not be polyhedral. A general proof of the theorem was given by Dinghass and extended by Herring,2 who also discussed some applications to liquid crystals. The proof is restated below in a concise mathematical form. Consider a body Po for which the surface tension y is a function of the direction only, i.e. y=y (n ) , and is defined as the specific surface free energy of any plane normal to n. If p is a point and rr a plane not containing p , rr divides the space into two half-spaces, one of which contains p . Define rr;(h,p) as the plane normal to n at a distance Ay(n) from p , where h B 0 is a scale factor. Let 8:; denote the half-space containing p together with all points in ~ ( h , p ) . Then --f