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Scanning specimens in liquids using commercial atomic force microscopy (AFM) is very time-consuming due to the necessary try-and-error iteration for determining appropriate triggering frequencies and probes. In addition, the iteration easily contaminates the AFM tip and damages the samples, which consumes probes. One reason for this could be inaccuracy in the resonant frequency in the feedback system setup. This paper proposes a frequency function which varies with the tip-sample separation, and it helps to improve the frequency shift in the current feedback system of commercial AFMs. The frequency function is a closed-form equation, which allows for easy calculation, as confirmed by experimental data. It comprises three physical effects: the quasi-static equilibrium condition, the atomic forces gradient effect, and hydrodynamic load effect. While each of these has previously been developed in separate studies, this is the first time their combination has been used to represent the complete frequency phenomenon. To avoid "jump to contact" issues, experiments often use probes with relatively stiffer cantilevers, which inevitably reduce the force sensitivity in sensing low atomic forces. The proposed frequency function can also predict jump to contact behavior and, thus, the probe sensitivity could be increased and soft probes could be widely used. Additionally, various tip height behaviors coupling with the atomic forces gradient and hydrodynamic effects are discussed in the context of carbon nanotube probes.

Frequency Function in Atomic Force Microscopy Applied to a Liquid Environment