The Atomic Force Microscope (AFM) scans the topography of a sample surface using a microsized flexible cantilever. In tapping-mode AFM, the tip-surface interactions are strongly nonlinear, rapidly changing and hysteretic. This paper explores, numerically, a flexible beam model that includes attractive, adhesive and repulsive contributions, as well as the interaction of the capillary fluid layers that cover both the tip and the sample in ambient conditions common in experiments. Forward-time simulation has been applied with an event handling numerical technique for dynamic analysis, and the Amplitude-Phase-Distance (APD) curves have been extracted. The branches of periodic solutions are found to end precisely where the cantilever comes into grazing contact with event surfaces in state space, corresponding to the onset of capillary interactions and the onset of repulsive forces associated with surface contact. The dissipated power, in the presence of conservative tip-sample interaction forces where the source of hysteresis is the formation and rupture of a liquid bridge between the tip and the sample, has been measured too. This simulation provides a more accurate way to validate the design of a new AFM probe and AFM controller than simulations which use the lumped-mass model.
One problem in imaging is due to coexistence of double stable responses which can reduce the precision of the images. Our target is comparing coexistence of double responses in liquid with responses obtained in air. Contact forces have some differences in liquid in comparison to air in magnitude and in the formulation. Hydrodynamic forces are also applied on the cantilever. This may change the nonlinear dynamics of tapping-mode Atomic Force Microscopy (AFM) in liquid in comparison to air. In this paper, we have simulated a tapping-mode AFM (TMAFM) cantilever in liquid environment and explored the existence of multi responses in amplitude and phase diagrams. For modeling we have used a continuous beam model with its first mode and forward-time simulation method for the solution of its hybrid dynamics. Frequency response results of the simulation show a good agreement with experiments. The results for studying the nonlinear dynamics of an AFM microcantilever show that while there are two stable responses in air, in liquid just one response exists. Amplitude diagrams show that there is not any shift from low amplitude to high amplitude and both diagrams of sweeping up and down of the driving frequency coincide on each other. Average force diagrams show that in liquid there is not any repulsive regime and the magnitude of attractive force is small too. Sensitivity analysis for studying the effect of the free amplitude shows that in common setpoints double responses do not occur in liquid. Frequency response results show that just in driving frequencies larger than the resonance frequency of the microcantilever in liquid double responses can occur.
In this article, the tip-sample interaction regimes in the presence of hysteretic forces are investigated using atomic force microscopy in the tapping mode. For this purpose, two samples that cause the formation of hysteretic forces, namely, silicon (stiff sample) with an adsorbed water film and polyethylene (compliant sample), are used. Also, for deriving the equation of motion of the microcantilever, the continuous beam model is used, and for determining the contact forces, depending on the sample under investigation, the Derjaguin–Muller–Toporov and Johnson–Kendall–Roberts contact mechanics models are used. The results indicate that the hysteretic interaction forces generate high-periodic and irregular responses at certain tip-sample separation distances. In fact, at these distances, a family of steady-state attractors is found that can be observed in one branch on the minimum tip-sample separation curves and in two separate branches on the average force curves. The reason for this occurrence might be the alternate formation of a liquid column between the probe tip and the sample (in the presence of ambient moisture), and for the compliant sample, the reason might be the alternate formation of an adhesion neck. In this article, the role of hysteretic forces in producing the hysteresis of the amplitude-separation curves is also explored.
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