Atomic force microscope cantilever spring constant evaluation for higher mode oscillations: a kinetostatic method

Our previous study of the particle mass sensor has shown a large ratio (up to thousands) between the spring constants of a rectangular cantilever in higher mode vibration and at the static bending or natural mode vibration. This has been proven by us through the derived nodal point position equation. That solution is good for a cantilever with the free end in noncontact regime and the probe shifted from the end to an effective section and contacting a soft object. Our further research shows that the same nodal position equation with the proper frequency equations may be used for the same spring constant ratio estimation if the vibrating at higher mode cantilever's free end has a significant additional mass clamped to it or that end is in permanent contact with an elastic or hard measurand object (reference cantilever). However, in the latter case, the spring constant ratio is much smaller (in tens) than in other mentioned cases at equal higher (up to fourth) vibration modes. We also present the spring constant ratio for a vibrating at higher eigenmode V-shaped cantilever, which is now in wide use for atomic force microscopy. The received results on the spring constant ratio are in good (within a few percent) agreement with the theoretical and experimental data published by other researchers. The knowledge of a possible spring constant transformation is important for the proper calibration and use of an atomic force microscope with vibrating cantilever in the higher eigenmodes for measurement and imaging with enlarged resolution.