Frequency shifts and analysis of AFM accompanying with coupled flexural–torsional motions

In a conventional dynamic atomic force microscopy (AFM), observing the flexural characteristics of a cantilever subjected to the tip–sample interaction is for extracting the topography and the material properties of a sample’s surface. Recently, Sahin et al. (2007) found that it is essential for understanding surface properties to design a cantilever with an eccentric tip and observe its coupled flexural–torsional characteristics. For effectively analyzing the flexural and torsional signals simultaneously, one has to find out the mode of a cantilever that the ratio of the tip gradient of flexural deformation and the tip torsional angle is comparable. Moreover, the development of an analytical model that can accurately simulate the surface-coupled dynamics of the cantilever is important for quantitative and qualitative understanding of measured results. In this paper, an analytical model of a cantilever with an eccentric tip and subjected to a nonlinear tip–sample force is established. The analytical solution is derived. It is found that the first two modes are the flexural motion and the third mode is the coupled flexural–torsional motion. Finally, the influences of several parameters on the tip angle ratio and frequency shift are investigated.