Thermal fuctuations and dynamic modelling of a dAFM cantilever

We discuss the Brownian thermal noise which affects the cantilever dynamics of a dAFM (dynamic atomic force microscope), both when it works in air and in presence of water. Our scope is to accurately describe the cantilever dynamics, and to get this result we deeply investigate the relationship between the cantilever thermal fluctuations and its interactions with the surrounding liquid. We present a relatively simple and very easy-to-use analytical model to describe the interaction forces between the liquid and the cantilever. The novelty of this approach is that, under the assumption of small cantilever oscillations, by using the superposition principle we found a very simple integral expression to describe fluid-cantilever interactions. More specifically we note that, beside including fluid inertia and viscosity (which is common to many existing models in the literature) an additional diffisivity term needs to be considered, whose crucial influence for the correct evaluation of the cantilever response to the thermal excitation is shown in the present paper. The coefficients of our model are obtained by using numerical results for a 2D fluid flow around a vibrating rectangular cross-section, and depend on the distance from the wall. This allowed us to completely characterize the dynamics of a dAFM cantilever also when it operates in tilted conditions. We validate the analytical model by comparing our results with numerical and experimental dAFM data previously presented in literature, and with experiments carried out by ourselves. We show that we can provide extremely accurate prediction of the beam response up and beyond the second resonant peak.