Nonlinear dynamics of atomic-force-microscope probes driven in Lennard-Jones potentials

The near-resonant, nonlinear dynamic response of microcantilevers in atomic force microscopy is investigated through numerical continuation techniques and simulations of discretized models of the microcantilever interacting with a surface through a Lennard-Jones potential. The tapping-mode responses of two representative systems, namely a soft silicon probe-silicon sample system and a stiff silicon probe-polystyrene sample system, are studied. Van der Waals interactions are shown to lead to a softening nonlinearity of the periodic solution response, while the short-range repulsive interactions lead to an overall hardening nonlinear response. Depending on the tip-sample properties, the dynamics of the microcantilevers occur either in asymmetric single-well potential regions or in asymmetric double-well potential regions. In both cases, forced periodic motions of the probe tip destabilize through a sequence of period-doubling bifurcations, while, in the latter, the tip can also escape the potential well to execute complex and unpredictable cross-well dynamics. The results predict a broad range of nonlinear dynamic phenomena, many of which have been observed in the literature on experimental atomic force microscopy.