Hydrodynamic interaction of AFM cantilevers with solid walls: An investigation based on AFM noise analysis

The noise power spectrum of the thermally activated motion of an AFM cantilever has been analyzed with respect to viscoelastic and hydrodynamic coupling between the cantilever and a substrate surface. Spheres with radii between 5 and 25 mum were glued to the cantilever to provide a well-defined geometry. The cantilever is modeled as a harmonic resonator with a frequency-dependent complex drag coefficient xi(omega). The variation of the drag coefficient xi(omega) with the tip-sample distance, D, and the sphere radius, R, can be expressed as a function of the single dimensionless parameter D/R. However, this scaling breaks down close to the surface. There are two sources of a frequency dependence of xi(omega), which are viscoelastic memory and hydrodynamics. Viscoelastic relaxation is observed when the surface is covered with a soft polymer layer. In the absence of such a soft layer one still finds a frequency dependence of xi(omega) which is caused by hydrodynamics. At large substrate-cantilever distances, the drag coefficient increases with frequency because,of inertial effects. At small distances, on the other hand, the drag coefficient decreases with increasing frequency, which is explained by the reflection of shear waves from the substrate surface. In liquids, inertial effects can be important when performing dynamic AFM experiments.