Analytical solution for size-dependent nonlinear behavior of AFM microcantilever with assembled probe in liquid environments

Abstract

Using modified strain gradient theory, this study aims to examine the size-dependent nonlinear behavior of atomic force microscopy (AFM) with an assembled cantilever probe (ACP) in various liquid environments. The largest contribution of this theory lies in its consideration of the material length scale parameter (MLSP) to analyze the effect of size on physical characteristics. When studying ACP in liquids, the hydrodynamic force creates a different behavior compared to an ACP beam enclosed in air. In this study, Hamilton’s principle is used to derive the nonlinear partial differential equation (PDE) of motion and associated boundary conditions (BCs) of the ACP in liquid environments. To achieve this, a combination of Euler-Bernoulli’s theory and strain gradient concepts are employed. As the next step, a reduced order model of the system is developed using the Galerkin method. To assess the nonlinear frequency response of the ACP in various liquid environments, perturbation technique, including the method of multiple scales (MMS), is applied to the developed reduced order model. The study also calculates resonance frequencies, phase planes, stability, and the potential function of the system analytically. The results of this study indicate that the system becomes nonlinear as the ratio of beam thickness to MLSP decreases. Several factors, including the ratio of beam thickness to MLSP as well as the viscous damping coefficient, can influence the jump phenomenon. Furthermore, the results of the phase and amplitude of the ACP in liquid environments demonstrate that an increase in the (h/l) ratio results in a decrease in response amplitude and a leftward shift in the phase of the response diagram. This phenomenon is indicative of a softening behavior exhibited by the system. Lastly, the paper examines the validity and accuracy of the MMS solution by comparing it to a numerical solution.