Accuracy of AFM cantilever thermal calibration in fluid of varying viscosities

Thermal calibration of atomic force microscopy (AFM) cantilever spring constant, k, is routinely performed by fitting the thermal displacement power spectral density (PSD) to a simple harmonic oscillator (SHO) model. Experiments performed in liquid present the choice of whether to also calibrate in liquid or to pre-calibrate in air, where reduced hydrodynamic drag produces a sharper resonance peak. It remains an open question whether this yields a more accurate calibration. Here we use a hydrodynamic model introduced by Sader and later extended by Bellon to generate thermal PSDs for two cantilevers of differing known stiffness across four media of increasing viscosity. We then fit each generated PSD with the SHO model used in experimental calibration (under various parameters and conditions). Iteratively fitting in the vicinity of the resonance peak, with logarithmic weighting and a fitted noise term, recovers the known spring constant to within 3% in every medium including the most viscous, where the fundamental mode of the softer cantilever is overdamped. We use this analysis to identify an optimal fitting strategy-in terms of window size, window center, and weighting-for experimentally reasonable cantilever properties and describe a general procedure for determining these parameters for other cantilevers. We show that the optimal fit-window width is not universal but can be determined for any cantilever and medium by applying the same analysis with different parameters. To confirm these results, we experimentally calibrated the same two types of cantilevers in air and liquid and obtained k values that agreed closely, despite several-fold changes in resonance frequency and quality factor upon immersion. These results demonstrate that AFM cantilevers intended for liquid-phase measurements can be reliably calibrated in situ, without recourse to a separate calibration in air.