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Effect of sidewall on frequency response and flexural sensitivity of AFM cantilevers immersed in liquid based on modified couple stress theory

In this paper, the flexural resonant frequency and flexural sensitivity of an atomic force microscope (AFM) with an assembled cantilever probe (ACP) immersed in a liquid environment are analyzed utilizing the modified couple stress (MCS) theory. The ACP including a horizontal cantilever, a vertical extension, and a tip fixed at the free end of the extension. For this aim, the governing partial differential equation (PDE) of motion and corresponding boundary condition is derived based on the Euler- Bernoulli theory and considering hydrodynamic functions in a liquid environment by employing the Hamilton principle. Then, according to this expression, the effect of the surface contact stiffness and the geometrical parameters such as ratio of beam thickness to the material length scale parameter and length of sidewall on the flexural resonance frequency and flexural sensitivity of ACP in liquid is assessed and compared with the case that the ACP works in the air environment. The result shows that the low-order vibration modes are more sensitive for low surface contact stiffness, so that, the best image contrast is obtained by excitation the first mode, but the situation is reversed when surface contact stiffness increasing. Also, the figures show that in contrast with the air environment, when the length of the assembled sidewall is smaller than one-half Quarter of the length of the horizontal cantilever, the assembled sidewall has no considerable impact on flexural sensitivity of ACP in the liquid environment.