Casimir and van der Waals forces near gently curved surfaces

The proximity force approximation (PFA) relates the interaction between closely spaced, smoothly curved objects to the force between parallel plates. We use a derivative expansion for gently curved surfaces to derive the leading curvature modifications to the PFA for the Casimir force. This approach can also be used to compute the leading and next-to-leading curvature corrections to the Casimir-Polder interaction between a polarizable small particle and a non-planar surface. For anisotropic nanoparticles the force depends on the orientation with respect to the surface; with the preferred orientation a function of curvature, temperature, as well as material properties. We also study the shift of rotational levels of a diatomic polar molecule due to its van der Waals (vdW) interaction with a gently curved dielectric surface at submicron separations. The level shifts, given by the quantum state averaged classical electrostatic interaction of the dipole with its image on the surface, depend on surface curvature and can be estimated by the derivative expansion. We argue that this curvature induced line splitting is experimentally observable.