The effect of hydrodynamic loading on the eigenmode shapes, modal stiffnesses, and optical lever sensitivities of atomic force microscope (AFM) microcantilevers is investigated by measuring the vibrations of such microcantilevers in air and water using a scanning laser Doppler vibrometer. It is found that for rectangular tipless microcantilevers, the measured fundamental and higher eigenmodes and their equivalent stiffnesses are nearly identical in air and in water. However, for microcantilevers with a tip mass or for picket shaped cantilevers, there is a marked difference in the second (and higher) eigenmode shapes between air and water that leads to a large decrease in their modal stiffness in water as compared to air as well as a decrease in their optical lever sensitivity. These results are explained in terms of hydrodynamic interactions of microcantilevers with nonuniform mass distribution. The results clearly demonstrate that tip mass and hydrodynamic loading must be taken into account in stiffness calibration and optical lever sensitivity calibration while using higher-order eigenmodes in dynamic AFM.
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The cantilevers were completely immersed in water several millimeters from any wall; thus squeeze film effects are negligible.
The lsqcurvefit command in MATLAB was used for the curve fitting. The first ODS was fitted to the function by choosing values of and , where is an arbitrary scaling constant. The second ODS was fitted to the function by considering to be fixed and equal to the value from the first ODS fit and choosing values of , , and .
Note that is assumed to have no dependence. This may not be valid if the microcantilever is close to and inclined with respect to the substrate (Ref. 52).
