Re-evaluation of the widely applied force-frequency relation for frequency-modulation AFM under solution

Frequency-modulation atomic force microscopy （FM-AFM） is a highly versatile tool for surface science. Besides imaging surfaces, FM-AFM is capable of measuring interactions between the AFM probe and the surface with high sensitivity, which can provide chemical information at sub-nanometer resolution. This is achieved by deconvoluting the frequency shift, which is directly measured in experiments, into the force between the probe and sample. At present, the widely used method to perform this deconvolution has been shown to be accurate under high quality （high-Q） factor vacuum conditions. However, under low quality （low-Q） factor conditions, such as in solution, it is not clear if this method is valid. A previous study apparently verified this relation for experiments in solution by comparing the force calculated by this equation with that obtained in separate experiments using the surface force apparatus （SFA）. Here we show that, in solution, a more direct comparison of the force calculated by this relation with that directly measured by the cantilever deflection in AFM reveals significant differences, both qualitative and quantitative. However, we also find that there are complications that hinder this comparison. Namely, while contact with the surface is clear in the direct measurements （including the SFA data）, it is less certain in the FM-AFM case. Hence, it is not clear if the two methods are measuring the same tip-sample distance regimes. Thus, our results suggest that a more thorough verification of this relation is required, as application of this formulation for experiments in solution may not be valid.