Gibbs-Wulff-Kaischew theorem

For the equilibrium form, the Gibbs-Wulff-Kaischew theorem can be applied. It states that the normal distances hi of all faces from the Wulff point, including the top face and the contact face /, are proportional to the specific surface energies (cf. eq. (4.10)). Hence, the specific surface energies in eq. (4.21b) can be replaced by the respective distances, hi/A = ov, so that... 

Figure 19. Illustration of the Gibbs-Wulff-Kaischew theorem, o-i to as are equal. The faces 1 to 6 are equidistant from the Wulff point. In the presence of the substrate, the distance h to the contact plane is reduced as indicated, becoming negative as ag >0-4.

This is the Gibbs-Wulff theorem as generalized by Kaischew for a crystal in contact with a substrate, i.e., for the case of heterogeneous nucleation and growth. Relation (4.10) allows the construction of the equilibrium form of a crystal ... 

Making use of Eqs. (2) and (3) the condition for the equilibrium form dO = 0, dV = 0 leads to Wulff s rule (Gibbs-Curie-Wulff theorem [iii-v]) generalized by - Kaischew [i] to account for the crystal-substrate interaction ... 

---