Analytical model of the nonlinear dynamics of cantilever tip-sample surface interactions for various acoustic atomic force microscopies

An analytical model is developed of the interaction of the cantilever tip of an atomic force microscope with the sample surface that treats the cantilever and sample as independent systems coupled by a nonlinear force acting between the cantilever tip and a volume element of the sample surface. To maintain equilibrium, the volume element is subjected to a restoring force from the remainder of the sample. The model accounts for the positions on the cantilever of the cantilever tip, laser probe, and excitation force (if any). The model leads to a pair of coupled nonlinear differential equations that are solved analytically using a matrix iteration procedure. Solutions are obtained for the phase and amplitude signals generated by various acoustic atomic force microscope (A-AFM) techniques including force modulation microscopy, atomic force acoustic microscopy, ultrasonic force microscopy, heterodyne force microscopy, resonant difference-frequency atomic force ultrasonic microscopy (RDF-AFUM), and amplitude modulation-atomic force microscopy (AM-AFM) (intermittent contact mode). The solutions are used to obtain a quantitative measure of A-AFM image contrast resulting from variations in the Young modulus of the sample. Applications of the model to measurements of LaRC (TM)-CP2 polyimide film using RDF-AFUM and AM-AFM images predict maximum variations in the Young modulus of 24% and 18%, respectively, over a common scan area. Both predictions are in good agreement with the value of 21% obtained from independent mechanical stretching measurements of the polyimide sheet material.