The exact solution of the Koga–Widom–Indekeu model and related models of wetting in fluid mixtures

We show how a broad class of two-component square-gradient models of wetting may be solved exactly for the surface tensions and density profile paths, and clarify how the presence or absence of critical point wetting, in binary and ternary mixtures, is related to universality and symmetry principles at critical end points. We begin by solving a model of fluid interfaces, first introduced by Koga and Widom, in ternary mixtures showing three phase coexistence. Numerical studies had revealed interesting wetting transitions, as well as curious geometrical properties of the profile paths in the density plane, and led these authors to conjecture expressions for the surface tensions. These conjectures were extended by Koga and Indekeu and predicted that partial wetting may persist up to the line of critical end points, i.e. critical point wetting was, unexpectedly, absent. Here, we obtain the exact density profiles and surface tensions for the Koga–Widom–Indekeu (KWI) model using complex analysis and drawing on the theory of algebraic curves. The exact solution determines the location and order of wetting transitions in the surface phase diagram, confirming that critical point wetting is absent. The model also displays the remarkable property that microscopic density profiles are mapped, by a conformal transform, onto the shape of a macroscopic drop near the contact line whose tensions satisfy the Neumann triangle. Two related models, which illustrate the role of the component isotropy, are also discussed: First, wetting by a critical layer where the critical singularities reflect an XY-like Casimir interaction mediated by the wetting film. Secondly, wetting in a binary mixture near a wall where critical point wetting is also absent and component density profiles can be obtained exactly revealing unexpected symmetries and connections to the mesoscopic disjoining pressure. These models suggest that a universality principle governs wetting in fluid mixtures, resolving contradicting results from earlier studies: Critical point wetting is present if the order-parameter components of the mixture describe Ising-like criticality, but is absent if there is a local XY symmetry. Implications for wetting transitions in more microscopic models and in experiments are discussed.