Statistical mechanics of warped membranes

Quenched random disorder can dramatically impact the mechanical properties of membranes through the renormalization of elastic moduli. We study an experimentally realizable class of disordered membranes with a uniform thickness, called warped membranes, whose shape is described by a random height profile h(x,y), which is characterized with a power spectral density P(q) that scales as a power law of the wavevector q: P(q) ~ q^-d_h. We fabricated such membranes by depositing aluminum oxide nanomembranes on rough surfaces with d_h=4 and then measured their effective bending rigidities. Upon varying the size and thickness of nanomembranes, we found that the average bending rigidity is enhanced, scaling with the system size and thickness in a manner not captured by conventional elasticity theory. Furthermore, sample-to-sample fluctuations in the bending rigidity also scale with system size and thickness, and the bending rigidity is not self-averaging. These scaling properties can be explained using a theoretical model that captures the coupling between the disorder and both tangential and normal deformations of the warped membrane. Warped membranes demonstrate how quenched disorder can be harnessed to control the mechanical properties of nanoscale structures.