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The theoretical foundations of dynamic atomic force microscopy (AFM) are based on point-mass models of continuous, micromechanical oscillators with nanoscale tips that probe local tip-sample interaction forces. In this letter, the authors present the conditions necessary for a continuous AFM probe to be faithfully represented as a point-mass model, and derive the equivalent point-mass model for a general eigenmode of arbitrarily shaped AFM probes based on the equivalence of kinetic, strain, and tip-sample interaction energies. They also demonstrate that common formulas in dynamic AFM change significantly when these models are used in place of the traditional ad hoc point-mass models.

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We study the physical origins of phase contrast in dynamic atomic force microscopy (dAFM) in liquids where low-stiffness microcantilever probes are often used for nanoscale imaging of soft biological samples with gentle forces. Under these conditions, we show that the phase contrast derives primarily from a unique energy flow channel that opens up in liquids due to the momentary excitation of higher eigenmodes. Contrary to the common assumption, phase-contrast images in liquids using soft microcantilevers are often maps of short-range conservative interactions, such as local elastic response, rather than tip-sample dissipation. The theory is used to demonstrate variations in local elasticity of purple membrane and bacteriophage φ29 virions in buffer solutions using the phase-contrast images.atomic force microscopy | liquid environments | energy dissipation | higher eigenmodes | momentary excitation D ynamic atomic force microscopy (dAFM) is an essential experimental tool for the study of conservative and dissipative forces on surfaces at nanometer length scales, which has major implications for the physics of biomolecular interactions, chemical bond kinetics, adhesion, wetting and capillary action, friction and elasticity on material surfaces (1, 2). dAFM techniques have been developed to distinguish between dissipative (friction, viscoelastic, bond breaking, surface hysteresis, capillary condensation) and conservative (elastic, magnetic, electrostatic) forces between a sharp oscillating tip and the surface. In amplitude-modulated AFM (AM-AFM) phase-contrast imaging, the variation in the phase of the oscillating probe tip with respect to the drive signal is mapped over the sample. For the past decade phase contrast has been intimately connected with variation in tip-sample dissipation over the sample (2-6). Phase-contrast imaging is widely recognized as perhaps the most important AM-AFM mode for the measurement of compositional contrast.The connection between phase contrast and tip-sample dissipation rests on the assumption that the cantilever dynamics can be modeled by a single eigenmode (point-mass) oscillator. With this assumption, the tip-sample dissipation can be equated to the difference between work input to the oscillator and energy dissipated into a surrounding viscous medium (4, 5). This theory forms the bedrock upon which phase-contrast imaging is currently based, at least under ambient and vacuum conditions. dAFM is now a well-known and broadly extended technique for nanoscale imaging and force spectroscopy in the biology community (7-10). Because the natural medium for the study of biological samples is liquid, it is of fundamental importance to develop a proper description of the different working modes of dAFM when the probe and sample are immersed in liquids. In particular, there is little work on understanding the origins of phase contrast in liquids (6) where soft cantilevers (stiffness 1 N/m) with low-quality factors (Q 5) are routinely used for the imaging of soft biological samples. It is im...

Standard spring constant calibration methods are compared when applied to higher eigenmodes of cantilevers used in dynamic atomic force microscopy (dAFM). Analysis shows that Sader's original method (Sader et al 1999 Rev. Sci. Instrum. 70 3967-9), which relies on a priori knowledge of the eigenmode shape, is poorly suited for the calibration of higher eigenmodes. On the other hand, the thermal noise method (Hutter and Bechhoefer 1993 Rev. Sci. Instrum. 64 1868-73) does not require knowledge of the eigenmode and remains valid for higher eigenmodes of the dAFM probe. Experimental measurements of thermal vibrations in air for three representative cantilevers are provided to support the theoretical results.