Hydrodynamic damping of a magnetically oscillated cantilever close to a surface

We studied the frequency response of a magnetically driven atomic force microscope (AFM) cantilever close to a sample surface in liquids. Amplitude–frequency (tuning) curves showed pronounced differences in dependence on the tip–sample separation (from 1 to 50 μm), with significant shifts of the resonance peak. A model was developed in which the cantilever was described in a full shape manner and the hydrodynamic forces acting on the cantilever were approximately calculated. The slight inclination of the cantilever to the surface (α∼15°) leads to a force profile along the cantilever. Therefore, the mathematical problem can be strictly solved only numerically. For an approximate analytical solution, the hydrodynamic force profile was approximated by a constant force along the cantilever for large separations and by a point force acting on the tip of the cantilever for small separations. The theoretical results calculated within this model agreed well with the experimental data and allowed to determine the cantilever mass in liquid M*, the joint mass at the tip end mt*, and the coefficient of viscous interaction of the cantilever with free liquid, γ∞.