Nanoparticle Morphology

Equilibrium Shape of a Macroscopic Crystal

   The shape of a crystal generally depends on growth conditions, which are usually very far from equilibrium, and for this reason it is not unique. However, under under conditions of thermodynamic equilibrium, the shape of a crystal is unique. This last result was first obtained by Wulff over a century ago [1]. The solution to this problem consists in minimising the total surface energy Es. For a liquid the result is immediate: one obtains a sphere. For a crystal, the specific surface energy γ depends on the orientation of the crystal face. One must therefore minimize

Fig. 1. Wulff construction of the equilibrium shape of a crystal from the γ-graph (dashed curve). O is the center of the crystal. The hexagon represents a projection of the equilibrium shape of the crystal (Wulff polyhedron).

where the index i represents the different facets with areas Ai and specific surface energy γi. Wulff showed that the minimal energy is obtained for a polyhedron in which the central distances hi to the faces are proportional to their surface energies γi. This is the well-known Wulff theorem:

 Using this theorem, if we know the dependence of the surface energy on the orientation, we can easily construct the equilibrium shape of the crystal.

Reference 

1. G. Wulff: Z. Krist.34, 449 (1901)