Elastica solution for a nanotube formed by self-adhesion of a folded thin film

Schmidt and Eberl demonstrated the construction of tubes with submicron diameters by the method of folding thin solid films [Nature (London)NATUAS 410, 168 (2001)]. In their method, a thin film is folded 180° and brought into adhesive contact with itself. The resulting sealed loop forms a nanotube with the thickness of the tube walls equal to the thickness of the thin film. The calculation of the diameter of the tube and the shape of its cross section in equilibrium are the subjects of this study. The tube is modeled as a two-dimensional elastica when viewed in cross section, and adhesive behavior is governed by an energy release rate criterion. A numerical technique is used to find elastic equilibria for a large range of material parameters. With these solutions in hand, the problem of designing a nanotube becomes transparent. It is shown that one dimensionless parameter determines the diameter of the nanotube, while another fixes its shape. Each of these parameters is a ratio involving the material's mechanical properties and the film thickness. Before concluding, we verify our model by comparing its results with the experimental observations of Schmidt and Eberl, for their materials.