## 2 Dynamic Contact Models and Simulation
### Mechanical Models
Considering the lateral contact stiffness and viscous stiffness, the mechanical model is equivalent to the model shown in Figure 2b, where \(y(t)\) represents the motion trail of the effective centroid, which is coupled elastically to the body \(M\) by a spring with an equivalent spring constant \(k_{y}\) in the \(Y(X)\) direction, \(V_{s}\) represents the approaching speed of the sample, and \(a\) represents the intermolecular distance. Based on the model by [PERSON] [34], the dynamic analysis is presented after full consideration of the stress condition where the microsphere tip is in contact with the sample surface. In addition, according to the wetting phenomenon [35] in nanotribology and the continuum medium principle, the water film with a certain thickness (about several nanometers) may be formed in microscale within the contact area because of the air humidity and the hydrophilic nature of the sample surface. Therefore, the initial motion equation in the \(Y(X)\) direction is expressed as:  
\[m_{y}\ddot{y}(t)+\beta\gamma\dot{y}(t)+\mu^{\prime}y(t)=F_{\rm drive}-\frac{ Kr^{3}}{2R}-\tau_{c}A-V-2\pi r\gamma_{\rm LG}\alpha(\cos\theta_{1}+\cos \theta_{2}) \tag{2}\]  
where \(y(t)\) is the motion of the effective centroid, \(\gamma\) is the general damping coefficient (\(\beta\) is a constant, let \(\beta=1\)), \(\mu^{\prime}\) is the gradient of the general elastic restoring force (which is the equivalent elastic stiffness \(k_{y}\)), \(R\) is the microsphere radius, \(K\) is the equivalent elastic constant modulus associated with the material, and \(K=4/3\)\(E^{*}\), \(E^{*}\) is the equivalent [PERSON]'s modulus, \(r\) is the sphere-sample surface contact radius, \(\tau_{c}\) is the shear stress acting on the adhesive contact area \(A\), \(V\) indicates the van der Waals force in the \(Y\) direction, \(\gamma_{\rm LG}\) 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, and \(\alpha\) is the coefficient related to the surface tension. In Equations (1) and (2), \(\beta\gamma\dot{z}(t)\) and \(\beta\gamma\dot{y}(t)\) represent the damping force (\(F_{\rm damp}\)), \(\mu z(t)\) and \(\mu^{\prime}y(t)\) represent the elastic restoring force (\(F_{\rm restore}\)), \(2\pi r\gamma_{\rm LG}\alpha(\cos\theta_{1}+\cos\theta_{2})\) is the capillary force (\(F_{\rm cap}\)), \(F_{att}=-HR/\big{[}6(z+z_{0})^{2}\big{]}\) is the long-range van der Waals force in the \(Y(X)\) direction, and \(F_{\rm ad}=-HR/\big{(}6a^{2}\big{)}\) and \(F_{\rm rep}=4/3E^{*}\sqrt{R}(a-z_{0}-z)^{3/2}\) are the adhesive force and short-range repulsive force, respectively. In Equation (2), \(F_{\rm ad}=S=\big{(}\mathrm{Kr}^{3}\big{)}/2R\) and \(F_{\rm fri}=\tau_{c}A\) indicate the adhesive force and interfacial friction force, respectively. The interfacial adhesive force is the dominant factor for interfacial friction during the dynamic contact process of the sphere-sample interaction in the friction model in the \(Y(X)\) direction; this force is generated by co-action of several microforces [30] and becomes the \"needle point force field\" within the nanocontact area. In the nanometer scale, the friction coefficient is no longer a constant value, and the relationship between interfacial friction (\(F_{\rm fri}\)) and normal pressure (\(F_{\rm N}\); namely, \(F_{\rm ad}\)) is nonlinear.