The Cahn–Hilliard residuum and its implications for critical-point wetting at solid solution surfaces

As part of their seminal 1958 analysis of nonuniform solutions, Cahn and Hilliard [J. Chem. Phys. 28, 258–267 (1958)] identified the Laplacian of the composition field as the key contribution to the excess free energy in solids with a conserved network of discrete atomic sites. Integration by parts converted to the now familiar gradient-square form underlying phase-field simulations and studies of critical-point wetting. The residuum from the integration by parts has since been ignored. Here, we examine its implications using critical-point wetting at the surface of an Ising-type solid solution for a representative case. As a benchmark, atomistic Monte Carlo simulation shows a continuous increase in wetting-layer thickness up to the bulk critical temperature. In contrast, the classical gradient-square continuum model predicts a first-order wetting transition. A more realistic Laplacian-based continuum formulation does not readily admit physically meaningful minimization by standard variational methods. As a discrete approach, a plane-by-plane model with an excess energy based on a second-difference (curvature) operator yields solutions consistent with the benchmark. Summation by parts transforms this operator into a first-difference- (gradient-) square form, and neglect of the residual term again leads to an incorrect first-order transition. Thus, the choice between curvature and gradient-square formulations alters the predicted character—first-order transition or continuous evolution in layer thickness—of the wetting, underscoring the fundamental importance of this choice.