  We consider a liquid drop sitting on a rough solid surface at equilibrium, a
volume constrained minimizer of the total interfacial energy. The large-scale
shape of such a drop strongly depends on the micro-structure of the solid
surface. Surface roughness enhances hydrophilicity and hydrophobicity
properties of the surface, altering the equilibrium contact angle between the
drop and the surface. Our goal is to understand the shape of the drop with
fixed small scale roughness. To achieve this, we develop a quantitative
description of the drop and its contact line in the context of periodic
homogenization theory, building on the qualitative theory of Alberti and
DeSimone.
