Shapes of grain inclusions in crystals

The ``equilibrium'' shape of a grain (with fixed volume) embedded in a simple-cubic crystal of the same material and rotated by a small angle with respect to the [001] axis is studied. The dislocation model of grain boundaries of Read and Schockley is used to compute the grain-boundary energy as a function of orientation. The grain shape is then found from this energy via the Wulff construction. Various aspects of the shape are analyzed for different values of both Poisson's ratio {ν} and a constant characterizing the core energy. One novel feature in this zero-temperature shape is curved portions which meet facets at edges where there is a discontinuity in slope. For small {ν} the shape is essentially a smoothly curved ellipsoid of revolution with four equivalent elliptical facets on the (100), (010), (1{}00), and (01{}0) planes. For large values of {ν} two smooth parts can meet at the equatorial plane with a slope discontinuity. In phase-transition language we find, besides first-order transitions, triple points and, in a narrow regime in {ν}, critical points. Effects of nonzero temperature and the dependence of the core energy on the character of the dislocation are explored qualitatively.