Elastic equilibrium of curved thin films

We present a unified theory of the bending of crystalline films that accounts for both elastic effects and crystal defects. Our theory predicts a transition from a bent coherent film with no dislocations to an incoherent, dislocated one as the film thickness or curvature is increased. The presence of the dislocations serves to renormalize the bending modulus of the system to smaller values. The degree to which the dislocations relax the elastic bending energy is found by calculating the equilibrium dislocation density and bending energy as a function of elastic constants, curvature, and film thickness. We demonstrate that at critical values of the curvature or thickness, there is a second-order phase transition between the undislocated and dislocated film. Generalizing these results to anisotropic elastic systems shows that weak bonding between crystal planes (such as in graphite) leads to a significant decrease in the critical curvature or thickness. An analysis of the case where the relaxation of the bending energy occurs by the formation of grain boundaries is also presented. We find that the introduction of grain boundaries can relieve the energy of the curved crystal more effectively than can the introduction of a uniform array of dislocations. Nonetheless, dislocation formation may be the dominant relaxation mechanism for very thin films (thin compared to the dislocation spacing in the grain boundary) and/or when dislocation migration kinetics are slow. Examples based upon nested fullerenes and bilayer surfactants are discussed.