Hydrodynamic Damping of Microcantilevers Oscillating Near Solid Walls at Low Knudsen Numbers

The vibrations of microcantilevers in Atomic Force Microscopes (AFM) or Radio Frequency (RF) switches strongly couple to the viscous hydrodynamics of the surrounding fluid in the vicinity of a solid wall. While prior efforts have focused on squeeze film damping effects at high Knudsen and squeeze numbers, the regime of low Knudsen and squeeze numbers is also very important for which squeeze film models need to be discarded in favor of unsteady Stokes hydrodynamics. We present convenient semi-analytical expressions for the unsteady viscous hydrodynamic functions of slender microbeams oscillating near solid walls in terms of key non-dimensional numbers. Using these expressions it becomes possible to predict semi-analytically the quality factors of multiple modes of microcantilever beams near solid walls in diverse applications ranging from AFM in liquids to RF microswitches under ambient conditions. The predictions compare favorably with fully three dimensional, transient flow-structure interaction computations as well as with preliminary experiments on cantilevers under ambient conditions.