Shape of a membrane on a liquid interface with arbitrary curvatures

TL;DR

This paper analyzes how a flat membrane deforms on liquid interfaces with various curvatures, deriving analytical solutions, confirming them numerically, and validating with experiments on polystyrene films.

Contribution

It provides the first analytical description of membrane deformation on arbitrary curved liquid interfaces, including cylindrical shapes and compression regions.

Findings

Membrane forms cylindrical shapes on negatively curved interfaces.

Outer regions compress azimuthally on positively curved interfaces.

Numerical and experimental results support the analytical predictions.

Abstract

We study the deformation of a liquid interface with arbitrary principal curvatures by a flat circular sheet. Working first at small slopes, we determine the shape of the sheet analytically in the membrane limit, where the sheet is inextensible yet free to bend and compress. We find that the sheet takes a cylindrical shape on interfaces with negative Gaussian curvature. On interfaces with positive Gaussian curvature, an inner region still adopts a cylindrical shape while the outer region is under azimuthal compression. Numerical energy minimization confirm our predictions and show that this behavior holds for finite slopes. Experiments on a thin polystyrene film at an anisotropic air-water interface show consistent behaviors.