Size dependent FSI model for tapping mode of atomic force microscopy cantilevers immersed in viscous fluid

In the tapping mode operation, atomic force microscopy (AFM) cantilevers are affected by a surrounding fluid medium and harmonic base excitation. This paper provides a size dependent fluid-structure interaction (FSI) model for presenting frequency response of AFM cantilevers excited around its resonance frequency. Based on the Navier–Stokes equation and the exponential decay assumption of velocity components, closed-form expression of pressure difference around an AFM cantilever moving in incompressible viscous flow is developed for the first time. The Galerkin method is used to solve the nonlocal couple stress Euler–Bernoulli beam equation including the effects of long-range cohesive interaction and microstructure local rotation. The closed-form solutions presented here for amplitude ratio of time-history response and quality factor can be used in the conceptual design of AFM cantilevers. For AFM cantilevers with low quality factor, there are significant errors for measuring driving forces, because the displacement of base is not negligible compared to the deflection of tip. The large differences are found to exist in the results of classical\(\:\:\)and modified couple stress theories and either different fluid mediums. Therefore, classical FSI model existing in the literature is inadequate for investigating performance of AFM cantilevers in force measuring and imaging modes.