## 2 Dynamic Contact Models and Simulation
### Mechanical Models
The tapping contact between the microsphere tip and sample surface is considered to be within the scope of the elastic contact. The mechanical model is equivalent to the mass-spring coupling system of the conventional optical lever model in AFM. In Figure 1b, \(z(t)\) represents the motion trail of the effective centroid, which is elastically coupled to the body M by a spring (equivalent spring constant \(k_{z}\)) in the \(Z\) direction. \(D\) represents the distance between the effective centroid and the sample surface (\(D=z+z_{0}\)), \(V_{s}\) represents the approaching speed of the sample, and \(a\) represents the intermolecular distance between the probe and the sample surface. Restoring force (\(F_{\mathrm{restore}}\)) is caused by the elastic deformation during the movement of the probe. Damping force (\(F_{\mathrm{damp}}\)) includes the air damping generated during the movement of the whole probe system and the energy dissipation in making contact with the sample surface. Tip-sample forces (\(F_{\mathrm{tip-sample}}\)) are those that exist between the microsphere tip and the sample, such as van der Waals force, capillary force, and so on. The arrows beside those forces in Figure 1b represent the direction of the forces. Both the approach and retract processes of the microsphere interacting with the sample in this mode are divided into two phases (i.e., \(D>a\) and \(D\leq a\)), because interfacial forces are associated with the distance between the microsphere tip and sample surface. In the \(D>a\) phase, the interfacial force is mainly represented by the long-range van der Waals attractive force (\(F_{\mathrm{att}}\)), whereas in the \(D\leq a\) phase, the interfacial force is mainly represented by surface adhesion (\(F_{\mathrm{ad}}\); Derjaguin-Muller-Toporov (DMT) theory) [33] and short-ranged repulsive force (\(F_{\mathrm{rep}}\)). Therefore, the initial motion equation in the \(Z\) direction is established as:  
\[\left\{ \begin{array}{ll}m_{z}\ddot{z}(t)+\beta\gamma\dot{z}(t)+\mu z(t)=F_{ \mathrm{drive}}-\frac{HR}{6\dot{z}(t+z_{0})^{2}}+2\pi r\gamma\gamma_{1}g(\cos \theta_{1}+\cos\theta_{2})&\text{for }D>a\\ m_{z}\ddot{z}(t)+\beta\gamma\dot{z}(t)+\mu z(t)=F_{\mathrm{drive}}-\frac{HR}{6 \dot{z}}+\frac{4}{3}E^{*}\sqrt{R}(a-z_{0}-z)^{3/2}+2\pi r\gamma_{1}g(\cos \theta_{1}+\cos\theta_{2})&\text{for }D\leq a\end{array} \right. \tag{1}\]  
where \(\gamma\) is the general damping coefficient (\(\beta\) is a constant, let \(\beta=1\)), \(\mu\) is the gradient of the general elastic restoring force (which is the equivalent elastic stiffness \(k_{z}\)), \(H\) is the Hamaker constant, \(R\) is the microsphere radius, \(z_{0}\) is the resonant center of the microsphere, \(z\) is the instantaneous tip position, \(r\) is the sphere-sample surface contact radius, \(\gamma_{1}\).g is the surface free energy of water (steam pressure), \(\theta_{1}\) is the contact angle between microsphere and water film, \(\theta_{2}\) is the contact angle between sample surface and water film, \(\alpha\) is the coefficient related to the surface tension, \(F_{\mathrm{drive}}\) is the driving force, and \(E^{*}\) is the effective Young's modulus. In the \(X\) and \(Y\) directions, such as the situation in the XOZ plane, the microsphere tip makes contact with the sample surface along the \(Y\) direction, as shown in Figure 2a. For convenience, the equivalent movement model in the XOZ plane is built by rotating the entire initial model by \(90^{\circ}\) counterclockwise, as shown in Figure 2b. According to nanotribology, adhesion and applied loads are in the same order of magnitude under nanofriction conditions. In other words, because of the existence of interfacial adhesion, generalized Hertz friction theory is not fully applicable to the present study.