Vibration of the cantilever in Force Modulation Microscopy analysis by a finite element model

Abstract

The resonance frequency of the cantilever beam of a Force Modulation Microscope is studied in function of the beam shape, the sample stiffness, and the contact model. We used a one dimensional finite element model for the cantilever beam, which permits the exact vibration of the beam to be treated in the contact mode, whatever its shape (rectangular as well as triangular beams) and excitation mode (by the beam holder, by the sample, by a localized, or distributed force). Based on a classic finite element scheme, it is simple to program and as rapid as the usual analytical models. We demonstrate that the mode of excitation of the beam strongly influences the cantilever’s frequency response in the contact mode. Anti-resonance is observed on the amplitude curves, which may perturb the measurements on some samples. We analyzed the true normal and tangential amplitude, for different beams and tip dimensions, in relation with the apparent amplitude of the vibration, as detected by the system. Experimental results on soft materials (polyurethane rubbers) give evidence of a nonsliding contact and underline the importance of an adequate treatment for the tangential forces, even in the linear domain. We assume a kind of viscosity of the contact, the effects of which are not visible for slow scan, but stick the tip and the sample together in the 10–100 kHz frequency range. This viscosity also induces an important damping of the resonance.