Abstract or Additional Information
We study the behaviour of a given volume of liquid confined between two rough solid plates. When the separation between the plates is small relative to the liquid volume, multiple capillary bridges are expected to form – depending on the surfaces' roughness – which minimise Gauss's capillary energy locally. To describe the geometry of the minimal configurations, we derive a $\Gamma$-expansion for the energy as the plate separation approaches zero, yielding a dimensionally reduced problem in terms of the wetted regions on the plates. At leading order, the energy is determined by the area of the wetted regions, while the second-order term, in the typical case, is given by their perimeter, weighted by an appropriate function of the relative adhesion coefficients and the limiting gap profile. This provides a framework for a successive phase-field approximation, which is employed in numerical simulations to study the evolution of the droplets under the normal and shear movement of the plates.
However, this expansion depends on the regularity of the leading-order $\Gamma$-limit's minimisers. When these sets are not of finite perimeter, a different scaling is required to capture the next-order correction, and the limit may not always exist. To illustrate this point, we construct minimising sets with energy scaling as sublinear powers of the plate separation parameter, depending on the Minkowski dimension of their boundary. This characteristic suggests there is no 'universal' scaling that encompasses all cases arising in this model.
This talk is based on joint work with Patrick Dondl, Lars Pastewka, and Yizhen Wang.