Analytical solutions of the frequency shifts of several modes in AFM scanning an inclined surface, subjected to the Lennard-Jones force

The role of higher cantilever modes is important to obtain some material contrast. The analysis of AFM subjected to a short-range force can improve greatly the studies of surface topography and interaction energies and interaction forces, especially for chemistry and biology materials. When the tip-sample distance is in the order of inter-atomic spacing, the short-range tip-sample force is usually simulated by the Lennard-Jones model. In typical AFM experiments the cantilever is inclined at an angle to the substrate surface. In this study, the analytical method to determine the frequency shift of AFM scanning the relative inclined surface, subjected to the Lennard-Jones force is proposed. The inclined angle may be the angle between the probe and the scanning table or due to the inclined plane and the roughness of sample surface. The closed-form solution of the partial differential equation with a nonlinear boundary condition is derived and then the corresponding frequency shifts of higher modes can be determined easily. The perturbation method transforming a continuous system into the lumped-masses one for determining the frequency shift will lead to an error. Although the above disadvantage exists, the lumped-masses model is simple and intuitive. Therefore, using the principle of dynamic strain energy, the conventional perturbation method is revised to determine the frequency shifts of higher modes in scanning an relative inclined surface. Further, the ratio relation among the frequency shifts is discovered. Moreover, it is found that the effect of the relative inclined angle on the first three frequency shifts is significant.