Free vibrations of U-shaped atomic force microscope probes

Abstract

Contact resonance atomic force microscope (AFM) methods have been used to quantify the elastic and viscoelastic properties of a variety of materials such as polymers, ceramics, biological materials, and metals with spatial resolution on the order of tens of nanometers. This approach involves measurement of the resonant frequencies of the AFM probe both for the free case and the case for which the tip is in contact with a sample. Vibration models of the probe and tip-sample contact models are then used to determine the sample properties from the frequency behavior and to create images of the sample properties. This work has been primarily focused on rectangular, single-beam probes for which the vibration models are relatively simple. Recently, U-shaped AFM probes have been developed to allow local heating of samples and the resonances of these probes are much more complex. In this article, a simplified analytical model of these U-shaped probes is described. This three beam model includes two beams clamped at one end and connected with a perpendicular cross beam at the other end. The beams are assumed only to bend in flexure and twist but their coupling allows a wide range of possible dynamic behavior. Results are presented for the first ten modes and the mode shapes are shown to have complex coupling between the flexure and twisting of the beams, particularly for the higher modes. All resonant frequency results are in good agreement with finite element results for the three probe designs and two values of thickness considered (all wavenumbers are within 3.0%). This work is anticipated to allow U-shaped probes to be used eventually for quantitative measurements of sample material properties during heating using a contact resonance approach.