Wetting, Algebraic Curves and Conformal Invariance

TL;DR

This paper provides an exact solution to a two-component fluid interface model, revealing conformally invariant algebraic curves for density profiles and confirming the absence of critical point wetting, with implications for surface tension and contact angle geometry.

Contribution

It introduces an exact analytical solution showing that density profiles are conformally invariant algebraic curves, clarifying geometric properties and surface tension behavior in the model.

Findings

Profile paths are conformally invariant quartic algebraic curves.

Critical point wetting is absent in the model.

Analytic representation of density profiles via complex conformal maps.

Abstract

Recent studies of wetting in a two-component square-gradient model of interfaces in a fluid mixture, showing three-phase bulk coexistence, have revealed some highly surprising features. Numerical results show that the density profile paths, which form a tricuspid shape in the density plane, have curious geometric properties, while conjectures for the analytical form of the surface tensions imply that nonwetting may persist up to the critical end points, contrary to the usual expectation of critical point wetting. Here, we solve the model exactly and show that the profile paths are conformally invariant quartic algebraic curves that change genus at the wetting transition. Being harmonic, the profile paths can be represented by an analytic function in the complex plane which then conformally maps the paths onto straight lines. Using this, we derive the conjectured form of the surface tensions and explain the geometrical properties of the tricuspid and its relation to the Neumann triangle for the contact angles. The exact solution confirms that critical point wetting is absent in this square-gradient model.