Nanomechanical measurements in Atomic Force Microscopy (AFM) are based on detection of quasi-static deflection (D) versus probe-sample vertical displacement (Z). These experimental D-vs-Z curves are converted into force (F) versus deformation (h) dependencies: F-vs-h. Elastic modulus and work of adhesion are extracted from analysis of F-vs-h curves in framework of solid state deformation models. Studies of D-vs-Z curves are associated with contact and non-resonant oscillatory modes of AFM. Dynamic probe variables, such as amplitude (A), phase (θ) and frequency shift (Δf), change as the oscillatory probe interacts with sample in the resonant oscillatory modes. The A-vs-Z, θ-vs-Z, Δf-vs-Z curves can be collected simultaneously in pairs if either frequency or phase is kept constant as it happens in amplitude modulation and frequency modulation imaging modes. A pair of (A-vs-Z, θ-vs-Z) curves is collected when the probe frequency is fixed at or near its resonance in free oscillatory state. A fixing of the probe phase at 90 degrees with a phase lock loop, which keeps the probe at effective resonance, allows recording a pair of (Δf-vs-Z, A-vs-Z curves). These curves are collected at single surface locations and their maps over the scanning area are often constructing to reflect spatial changes of materials’ properties. Both pairs of curves contain at least the same information as quasi-static D vs Z curve, and the latest can be potentially restored from one of the pairs. We use steady state equations of asymptotic amplitude-phase AFM dynamics derived using Krylov-Bogoliubov-Mitropolsky( KBM) averaging technique. This technique allows a formulation of the force restoration problem as a pair of integral equations. However, solution to these equations is an ill-posed inverse problem that magnifies noise and requires regularization. One form of regularization can use a priori parametric force model, e.g. JKR, but even in this case non-parametric restoration is required to justify the model.
We demonstrate a successful use of Tikhonov regularization for the force restoration from both pairs (A-vs-Z, θ-vs-Z) and (Δf-vs-Z, A-vs-Z) of curves by simulation with different kind of forces, errors and noise levels. The presented theoretical analysis was verified in interplay with experimental data, which were obtained on samples with macroscopic elastic modulus varied in the range form few MPa to tens of GPa. They included neat polymers, polymer blends, more complex organic compositions and metals. The AFM probes with spring constants from few to hundreds of N/m were applied in these studies.