Material property analytical relations for the case of an AFM probe tapping a viscoelastic surface containing multiple characteristic times

We explore the contact problem of a flat-end indenter penetrating intermittently a generalized viscoelastic surface, containing multiple characteristic times. This problem is especially relevant for nanoprobing of viscoelastic surfaces with the highly popular tapping-mode AFM imaging technique. By focusing on the material perspective and employing a rigorous rheological approach, we deliver analytical closed-form solutions that provide physical insight into the viscoelastic sources of repulsive forces, tip–sample dissipation and virial of the interaction. We also offer a systematic comparison to the well-established standard harmonic excitation, which is the case relevant for dynamic mechanical analysis (DMA) and for AFM techniques where tip–sample sinusoidal interaction is permanent. This comparison highlights the substantial complexity added by the intermittent-contact nature of the interaction, which precludes the derivation of straightforward equations as is the case for the well-known harmonic excitations. The derivations offered have been thoroughly validated through numerical simulations. Despite the complexities inherent to the intermittent-contact nature of the technique, the analytical findings highlight the potential feasibility of extracting meaningful viscoelastic properties with this imaging method.