It was recalled that large twist-angle grain boundaries in layered structures were often described by Scherk's first surface, whereas small twist-angle grain boundaries were usually described in terms of an array of screw dislocations. It was shown here that there was no essential difference between these 2 descriptions and that, in particular, their comparative energetics depended sensitively upon the core structure of their screw dislocation topological defects. Scherk's surface was an anisotropic dilatation of a periodic surface that was constructed from a single set of strength-2 screw dislocations. Since the lamellar ground state had a preferred layer spacing, layer compression contributed to the free energy of the structure. This broke the dual mapping between the helicoids. It followed that Scherk's surface was a twist grain boundary which was composed of a single set of parallel screw dislocations, and that the geometry of these defects created a perpendicular set of helicoidal structures in the surface. It was also demonstrated that Scherk's surface had a higher energy than a structure built up of +1 dislocations; for small angles. In the case of biphase materials, the +1 dislocations considered here would be topologically forbidden; they would become +1/2 dislocations. In this case, the energetic competition would be between Scherk's surface and an +2 structure of the above type. In either case, a detailed analysis of the core structure would be required in order to make an unambiguous prediction of the most stable structure at larger angles.

Minimal Surfaces, Screw Dislocations and Twist Grain Boundaries. R.D.Kamien, T.C.Lubensky: Physical Review Letters, 1999, 82[14], 2892-5