A spectral criterion for faceting at alloy solid-liquid interfaces

A long-standing question in solidification is which interfaces are atomically smooth and which are rough. The classical Jackson factor addresses this for pure substances via a single nondimensional ratio between latent heat and thermal energy. Here, I derive a generalization for an n -component alloy that remains algebraic yet captures coupling between species. Starting from a bond-counting free energy on a partially filled surface layer, I compute the Hessian at the “rough” coverage point and, after an exact concentration rescaling, obtain an n × n coupling matrix S . The interface is rough if the largest eigenvalue satisfies λ max ( S ) < 2; otherwise, it facets along the top eigenvector. This spectral criterion reduces to Jackson’s α < 2 for a single component, gives eigenvalue-free row-sum bounds, and provides closed forms for important symmetry classes (e.g., B2 {110}). Applications to Al-rich multicomponent alloys show how both microalloying and complex chemistries amplify λ max ( S ), rationalizing the observation that “complexity promotes faceting.”