Two methods commonly used to measure the normal spring constants of atomic force microscope cantilevers are the added mass method of Cleveland et al. [J. P. Cleveland et al., Rev. Sci. Instrum. 64, 403 (1993)], and the unloaded resonance technique of Sader et al. [J. E. Sader, J. W. M. Chon, and P. Mulvaney, Rev. Sci. Instrum. 70, 3967 (1999)]. The added mass method involves measuring the change in resonant frequency of the fundamental mode of vibration upon the addition of known masses to the free end of the cantilever. In contrast, the unloaded resonance technique requires measurement of the unloaded resonant frequency and quality factor of the fundamental mode of vibration, as well as knowledge of the plan view dimensions of the cantilever and properties of the fluid. In many applications, such as frictional force microscopy, the torsional spring constant is often required. Consequently, in this article, we extend both of these techniques to allow simultaneous calibration of both the normal and torsional spring constants. We also investigate the validity and applicability of the unloaded resonance method when a mass is attached to the free end of the cantilever due to its importance in practice.

The frequency response of a cantilever beam is strongly dependent on the fluid in which it is immersed. In a companion study, Sader [J. Appl. Phys. 84, 64, (1998)] presented a theoretical model for the flexural vibrational response of a cantilever beam, that is immersed in a viscous fluid, and excited by an arbitrary driving force. Due to its relevance to applications of the atomic force microscope (AFM), we extend the analysis of Sader to the related problem of torsional vibrations, and also consider the special case where the cantilever is excited by a thermal driving force. Since longitudinal deformations of AFM cantilevers are not measured normally, combination of the present theoretical model and that of the companion study enables the complete vibrational response of an AFM cantilever beam, that is immersed in a viscous fluid, to be calculated.

Theoretical models for the frequency response of a cantilever beam immersed in a viscous fluid commonly assume that the fluid is unbounded. Experimental measurements show, however, that proximity to a surface can significantly affect the frequency response of a cantilever beam. In this article, we rigorously calculate the effect of a nearby surface on the frequency response of a cantilever beam immersed in a viscous fluid, and present a general theoretical model. Due to its practical relevance to applications of the atomic force microscope and microelectromechanical systems, detailed results are presented for cantilever beams with rectangular geometries executing flexural and torsional oscillations. It is found that dissipative loading in the fluid is primarily responsible for the observed variation in the frequency response, whereas inertial loading exerts a relatively weak influence.

The hydrodynamic loading on a solid body moving in a viscous fluid can be strongly affected by its proximity to a surface. In this article, we calculate the hydrodynamic load on an infinitely long rigid beam of zero thickness that is undergoing small amplitude oscillations. The presence of a solid surface an arbitrary distance from the beam is rigorously accounted for using a boundary integral formulation.