Perforations, Curvature, and Thermal Fluctuations in Free-Standing Graphene
Abstract: Understanding deformations of macroscopic thin plates and shells has a long and rich history, culminating with the Foeppl-von Karman equations in 1904, characterized by a dimensionless coupling constant (the "Foeppl-von Karman number") that can easily reach vK = 107 in an ordinary sheet of writing paper. However, thermal fluctuations in thin elastic membranes fundamentally alter the long-wavelength physics. We discuss the remarkable properties of free-standing graphene sheets (with vK = 1013!) at room temperature, where enhancements of the bending rigidity by factors of ~4000 compared with T = 0 values have now been observed. Thermalized elastic membranes can undergo a crumpling transition when the microscopic bending stiffness is comparable to kT. We argue that the crumpling temperature can be dramatically reduced by inserting a regular lattice of laser-cut perforations. These expectations are confirmed by extensive molecular dynamics simulations, which also reveal a remarkable "frame crumpling transition" triggered by a simple large hole inserted into a graphene sheet. We show finally that thin amorphous spherical shells with a background Gaussian curvature are inevitably (in the absence of a stabilizing pressure difference) crushed by thermal fluctuations beyond a critical size, of order 160 nm for graphene at room temperature.