Transfer of Mechanical Vibration from a Sample to an AFM-Cantilever - A Theoretical Description

Quantitative evaluation of acoustic AFM images to determine elastic, anelastic, and adhesive surface properties require the theoretical description and calculation of the transfer of mechanical vibrations from an insonified sample to an AFM-cantilever if its sensor tip is in contact with the sample surface. This paper presents a theory about flexural vibrations excited in a rectangular beam cantilever by out-of-plane vibrations of the sample surface taking into account the non-linearity of the interaction forces between sensor tip and sample surface and without restriction to small amplitudes. From a general formal solution of the position and time dependent partial differential equation of flexural vibrations and the boundary conditions of the clamped surface-coupled cantilever a differential equation only in time of the cantilever deflection is analytically derived. The numerical solution of this equation yields the sensor tip motion including the transfered high-frequency vibrations as well as the mean cantilever deflection and the cantilever slope at the contact point. Numerical results for some examples are discussed, especially with regard to the potential of the measurement quantities for quantitative materials characterization.