Cahn-Hoffman theorem

The Cahn-Hoffman theorem describes how the shape of an interface, like a grain boundary or crystal surface, is determined by its surface energies. It states that the equilibrium shape minimizes total surface energy, leading to a stable form where the surface tension varies with orientation but still conforms to a specific geometric condition. Essentially, the theorem provides a mathematical rule showing how the geometry of a surface naturally adjusts to balance the energy distribution, resulting in predictable shapes like crystals or crystalline interfaces, emphasizing the link between microscopic energy differences and macroscopic form.