Long-range hydrodynamic forces in liquid FM-AFM

We study the effects of hydrodynamic forces in frequency-modulation AFM experiments (FM-AFM) in liquid. We first establish the theoretical equations needed to derive the interaction stiffness kint and the damping βint due to the hydrodynamic forces from the frequency shift and the excitation amplitude. We develop specific FM-AFM experiments to measure the variation of kint and βint over a large range of distance in water up to 200 µm. Comparison between theory and experiments point out that the evolution of kint at short and long distance arises from unsteady hydrodynamic forces on the cantilever. On the other hand, βint is small at long distance and diverges at short probe-surface distance, as predicted by the classical Reynolds sphere model.