Winterbottom Shape

A Winterbottom shape is a truncation of a Wulff shape by a half-space, followed by translation and dilation, that minimizes anisotropic surface energy with a flat substrate contact term at prescribed volume (Winterbottom 1967, Kreutz and Schmidt 2026). It is the geometric solution of the corresponding constrained problem in the calculus of variations.

Let be a positive continuous function on the unit sphere whose positively homogeneous extension is convex, and let be the difference between the crystal-substrate and substrate-vapor energies per unit area. For a region above the substrate , the energy is

where is the outward unit normal vector and denotes -dimensional area. The integrals are interpreted measure-theoretically for sets of finite perimeter.

Writing for the vertical unit vector, the partial-wetting range is . In this range the minimizing shape is

up to translation parallel to the substrate and sets of zero volume, where the positive scale factor fixes the prescribed volume (Kreutz and Schmidt 2026). The translation by places the cut on the substrate. The figure above illustrates the cut for before this translation.

When is constant, the Wulff shape is a closed ball and the Winterbottom shape is a spherical cap in three dimensions. The contact angle , measured through the drop, satisfies . Thus gives a hemisphere, while negative gives a flatter cap.