Difficulties in fitting the thermal response of atomic force microscope cantilevers for stiffness calibration

Abstract

This paper discusses the difficulties of calibrating atomic force microscope (AFM) cantilevers, in particular the effect calibrating under light fluid-loading (in air) and under heavy fluid-loading (in water) has on the ability to use thermal motion response to fit model parameters that are used to determine cantilever stiffness. For the light fluid-loading case, the resonant frequency and quality factor can easily be used to determine stiffness. The extension of this approach to the heavy fluid-loading case is troublesome due to the low quality factor (high damping) caused by fluid-loading. Simple calibration formulae are difficult to realize, and the best approach is often to curve-fit the thermal response, using the parameters of natural frequency and mass ratio so that the curve-fit's response is within some acceptable tolerance of the actual thermal response. The parameters can then be used to calculate the cantilever stiffness. However, the process of curve-fitting can lead to erroneous results unless suitable care is taken. A feedback model of the fluid–structure interaction between the unloaded cantilever and the hydrodynamic drag provides a framework for fitting a modeled thermal response to a measured response and for evaluating the parametric uncertainty of the fit. The cases of uncertainty in the natural frequency, the mass ratio, and combined uncertainty are presented and the implications for system identification and stiffness calibration using curve-fitting techniques are discussed. Finally, considerations and recommendations for the calibration of AFM cantilevers are given in light of the results of this paper.