Cantilever Dynamics: Theoretical Modeling

Abstract

To provide a measure of flexibility and symmetry regarding the description of tip-sample interactions, a dynamical model is presented for which the cantilever tip and the sample surface are treated as independently damped simple harmonic oscillators passively coupled via the nonlinear tip-sample interaction forces. The sample oscillations are assumed to occur in the coupling from a small element of surface mass (active mass) attached to the remainder of the sample for which the spring constant is the sample stiffness constant. The analytical model reduces to a pair of coupled nonlinear differential equations, the general solutions of which are obtained using a matrix iteration procedure. The general solutions are applied to the quantitative assessment of signal generation and contrast mechanisms in atomic force acoustic microscopy (AFAM), force modulation microscopy (FMM), ultrasonic force microscopy (UFM), ultrasonic atomic force microscopy (UAFM), amplitude modulation atomic force microscopy (AM-AFM), and scanning near-field ultrasonic holography (SNFUH) including the related heterodyne force atomic force microscopy (HF-AFM) and resonant difference-frequency atomic force ultrasonic microscopy (RDF-AFUM). In addition to obtaining quantitative expressions for surface contrast mechanisms, contrast mechanisms from subsurface features are accounted in the model for AFAM, FMM, UFM, and SNFUH.