Design of V-shaped cantilevers for enhanced multifrequency AFM measurements

Abstract

As the application of atomic force microscopy (AFM) in soft matter characterization has expanded, the use of different types of cantilevers for these studies have also increased. One of the most common types of cantilevers used in soft matter imaging is V-shaped cantilevers due to their low normal spring constant. These types of cantilevers are also suitable for nanomanipulation due to their high lateral spring constants. The combination of low normal spring constant and high lateral spring constants makes V-shaped cantilevers promising candidates for imaging soft matter. Although these cantilevers are widely used in the field, there are no studies on the static and dynamic behavior of V-shaped cantilevers in multifrequency AFM due to their complex geometry. In this work, the static and dynamic properties of V-shaped cantilevers are studied while investigating their performance in multifrequency AFM (specifically bimodal AFM). By modeling the cantilevers based on Timoshenko beam theory, the geometrical dimensions such as length, base width, leg width and thickness are studied. By finding the static properties (mass, spring constants) and dynamic properties (resonance frequencies and quality factors) for different geometrical dimensions, the optimum V-shaped cantilever that can provide the maximum phase contrast in bimodal AFM between gold (Au) and polystyrene (PS) is found. Based on this study, it is found that as the length of the cantilever increases the 2nd eigenmode phase contrast decreases. However, the base width exhibits the opposite relationship. It is also found that the leg width does not have a monotone relationship similar to length and base width. The phase contrast increases for the range of 14 to 32 µm but decreases afterwards. The thickness of a V-shaped cantilever does not play a major role in defining the dynamics of the cantilever compared to other parameters. This work shows that in order to maximize the phase contrast, the ratio of second to first eigenmode frequencies should be minimized and be close to a whole number. Additionally, since V-shaped cantilevers are mostly used for soft matter imaging, lower frequency ratios dictate lower spring constant ratios, which can be advantageous due to lower forces applied to the surface by the tip given a sufficiently high first eigenmode frequency. Finally, two commercially available V-shaped cantilevers are theoretically and experimentally benchmarked with an optimum rectangular cantilever. Two sets of bimodal AFM experiments are carried out on Au-PS and PS-LDPE (polystyrene and low-density polyethylene) samples to verify the simulation results.