Thermal Fluctuations and Dynamic Modeling of a dAFM Cantilever

Abstract

The Brownian thermal noise which affects the cantilever dynamics of a dynamic atomic force microscope (dAFM) is discussed, both when it works in air and in presence of water. The relationship between the cantilever thermal fluctuations and its interactions with the surrounding liquid is investigated, and an analytical model to describe the interaction forces is presented. The novelty of this approach is that a very simple integral expression to describe fluid-cantilever interactions is found. More specifically, besides including fluid inertia and viscosity, an additional diffusivity term needs to be considered, whose crucial influence for the correct evaluation of the cantilever response to the thermal excitation is shown. The coefficients of the model are obtained by using numerical results for a 2D fluid flow around a vibrating rectangular cross section, so that the dynamics of a dAFM cantilever can be characterized when it also operates in tilted conditions. The analytical model is validated by comparison with numerical and experimental data previously presented in literature, and with experiments carried out by the authors. It is shown that the present model can provide an extremely accurate prediction of the beam response up and beyond the second resonant peak.

A theoretical-experimental framework for the analysis of the dynamic response of a QEPAS tuning fork device immersed in a fluid medium

“…Equation (6), which satisfies the causality principle, clearly is an heuristic approximation of the fluid response. Furthermore, the dependence of the three fluid coefficients on the spatial abscissa x is negligible in the case of an isolated cantilever vibrating in an unbounded fluid [47], as in the case considered in the present study. To estimate the numerical values assumed by the three coefficients in our application, the best fit of the experimental response has to be performed.…”

“…Assuming small in-plane deflections, the fluid response can be considered linear and the force f F i x; t ð Þ can be evaluated by means of the following Green's function approach [47]…”

“…By relying on the linearity of system's Equations (12), a Green's function approach has been adopted to determine the specific solution of the problem. Hence, the system's complex Green's function is calculated, even referred to the susceptibility function, which is the solution of the so-called fundamental problem [47]. Specifically, a concentrated force of unit impulse is supposed to be applied to a generic cross-section n of the resonator (Figure 3), that is, to a certain section n of the j-th cantilever, when the other remains unloaded.…”

“…The system domain includes two Euler-Bernoulli beams partially coupled by a distributed linear spring [35], used to model the plate connecting the two prongs as well as the underlying part. The drag force exerted by the fluid on the QTF is modelled by a simple integral expression which accounts not only for fluid inertia and viscosity but also for a diffusivity vorticity term [42][43][44], whose influence is fundamental for the correct evaluation of the system response [47].…”