Euler-Bernoulli theory accurately predicts atomic force microscope cantilever shape during non-equilibrium snap-to-contact motion

Abstract

We prove that the Euler-Bernoulli elastic beam theory can be reliably used to describe the dynamics of an atomic force microscope cantilever during the far from equilibrium snap-to-contact event. In conventional atomic force microscope operation, force-separation curves are obtained by post-processing voltage versus time traces produced by measuring one point on the cantilever close to the hanging end. In this article, we assess the validity of the Euler-Bernoulli equation during the snap-to-contact event. The assessment is based on a direct comparison between experiment and theory. The experiment uses Doppler vibrometry to measure displacement versus time for many points along the long axis of the cantilever. The theoretical algorithm is based on a solution of the Euler-Bernoulli equation to obtain the full shape of the cantilever as a function of time. The algorithm uses as boundary conditions, experimentally obtained information only near the hanging end of the cantilever. The solution is obtained in a manner that takes into account non-equilibrium motion. Within experimental error, the theory agrees with experiment indicating that the Euler-Bernoulli theory is appropriate to predict the cantilever kinematics during snap-to-contact. Since forces on the tip can be obtained from the instantaneous shape of the cantilever, this work should allow for computation of tip-sample forces during the snap-to-contact event from a conventional force-distance measured input.