Reconstructing the distributed force on an atomic force microscope cantilever

A methodology is developed to reconstruct the force applied to an atomic force microscopy (AFM) cantilever given the shape in which it vibrates. This is accomplished by rewriting Bernoulli–Euler beam theory such that the force on the cantilever is approximated as a linear superposition of the theoretical cantilever eigenmodes. The weighting factors in this summation are calculated from the amplitude and phase measured along the length of the cantilever. The accuracy of the force reconstruction is shown to depend on the frequency at which the measurement is performed, the number of discrete points measured along the length of the cantilever, and the signal-to-noise ratio of the measured signal. In contrast to other AFM force reconstruction techniques, this method can reconstruct the distribution of force applied over the length of the AFM cantilever. However, this method performs poorly for localized forces applied to the cantilever, such as is typical of most tip–sample interaction forces. Proof of concept experiments are performed on an electrostatically excited cantilever and the expected force distribution is recovered. This force reconstruction technique offers previously unavailable insight into the distributed forces experienced by an AFM cantilever. (paper)