Geometry-driven self-assembly of interfacial sheets

When a linear elastic material is made sufficiently thin, the energy required to bend the sheet is orders of magnitude lower than the energy required to stretch it. Placing this sheet on a liquid interface introduces another energy scale. In the case of an inextensible sheet with zero bending cost – an asymptotic limit that is readily achieved in experiments using ultrathin polymer films – the sheet will bend and wrinkle in such a way as to minimize the exposed liquid surface area [1]. However, predicting the overall shape of the film is a nontrivial optimization problem that is highly sensitive to the curvature of the interface or the boundary conditions. We use Surface Evolver simulations and analytic calculations to study the energetic cost of placing an ultrathin elastic disc on an arbitrary local topography of moderate curvature, from spherical to ellipsoidal to saddle-shaped interfaces. We establish scaling laws that relate the system energy to the principal radii of curvature, which capture our measurements spanning nearly three decades in curvature. This work paves the way for designing curved interfacial topographies that promote self-assembly via energy minimization.
[1] Paulsen et al., Nat. Mater. 14 (2015).