The Atomic Force Microscope (AFM) has become an indispensable tool in biology because it permits the probing of nanomechanical properties under physiological (liquid environments) conditions. AFM has been used in liquid environments to image, manipulate and probe atoms, living cells, bacteria, viruses, subcellular structures such as microtubules, individual proteins and DNA. Probably the most popular method used for AFM in liquids is the tapping mode wherein a resonant micro-cantilever is scanned over a sample. Yet very little is known about the dynamics of micro-cantilevers in liquid environments while interacting with nanostructures. This problem is especially challenging because viscous hydrodynamics couples strongly with cantilever motions, and the contribution from the electric double layer forces, which is not significant in air, must be taken into account. Previous attempts in the analysis and simulation of tapping mode in liquid modeled the tip-sample interaction forces using either a Lennard-Jones potential [1, 2], an exponentially growing force of small duration of the cantilever oscillation cycle [3] without any contact mechanics, or an unrealistic discontinuous interaction force [4]. Moreover, in all these papers the micro-cantilever was modeled by a point (lumped) mass, and the hydrodynamic effects were not derived rationally from basic hydrodynamic theory. Instead, a low quality factor (Q factor) and an added fluid mass were simply assumed [1-4]. A direct method to systematically deal with the AFM micro-cantilever using continuous beam theory in liquids governed by the unsteady Stokes equations and experiencing intermittent contact with the sample is not available in the literature.
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