Boundary curvature effect on the wrinkling of thin suspended films

In this letter, we demonstrate a relation between the boundary curvature $\kappa$ and the wrinkle wavelength $\lambda$ of a thin suspended film under boundary confinement. Experiments are done with nanocrystalline diamond films of thickness $t \approx 184$~nm grown on glass substrates. By removing portions of the substrate after growth, suspended films with circular boundaries of radius $R$ ranging from approximately 30 to 811 $\mu$m are made. Due to residual stresses, the portions of film attached to the substrate are of compressive prestrain $\epsilon_0 \approx 11 \times 10^{-4}$ and the suspended portions of film are azimuthally wrinkled at their boundary. We find that $\lambda$ monotonically decreases with $\kappa$ and present a model predicting that $\lambda \propto t^{1/2}(\epsilon_0 + \Delta R \kappa)^{-1/4}$, where $\Delta R$ denotes a penetration depth over which strain relaxes at a boundary. This relation is in agreement with our experiments and may be adapted to other systems such as plant leaves. Also, we establish a novel method for measuring residual compressive strain in thin films.