Fig. 5: Potential function \(U\left( {{\Delta}\phi } \right)\) for the generalized theoretical consideration.

The AFM-cantilever translates \(U\left( {{\Delta}\phi } \right)\) in the ultra-light tapping mode operation. \(U\left( {{\Delta}\phi } \right)\) in the case of soft-matter is expected to typically exhibit two minima¹³ that correspond to steady-state and second metastable energy. This is dynamically captured by the cantilever as minima at \({\Delta}\phi _ +\) and \({\Delta}\phi _ -\) while translating between the states. The minima are separated with a potential barrier \({\Delta}E_{\rm{fluc}}\). It must be noted here, the potential barrier can consist of multiple transitional levels each having its own characteristics of relaxation timescale that would account for a phase delay \({\Delta}\phi\) proportional to the bandwidth product \({\Delta}_\omega \cdot \tau _c,\) where \({\Delta}_\omega\) is the frequency dispersion and \(\tau _c\) is the correlation time. Energy losses from fluctuations in the limit \({\Delta}\phi \to 0\) compensates the thermodynamic cost of transition to the metastable state \({\Delta}\phi _ -\) from the steady state \({\Delta}\phi _ +\).