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 k _int and the damping β _int due to the hydrodynamic forces from the frequency shift and the excitation amplitude. We develop specific FM-AFM experiments to measure the variation of k _int and β _int over a large range of distance in water up to 200 µm. Comparison between theory and experiments point out that the evolution of k _int at short and long distance arises from unsteady hydrodynamic forces on the cantilever. On the other hand, β _int is small at long distance and diverges at short probe-surface distance, as predicted by the classical Reynolds sphere model.