## 2 Dynamic Contact Models and Simulation
### Microforces and Mathematical Simulation  
As the main cause of interfacial friction, the interfacial adhesive force is resulted from the co-action of the van der Waals force, capillary force, and other microforces. These forces should be analyzed in order to understand the mechanical properties and triggering mechanism in three dimensions. The van der Waals force is a type of electromagnetic force that has neither directivity nor saturability, and has an effective range from nanometer to sub-nanometer. It mainly arises from three kinds of forces with different generating mechanisms: orientation force between polar molecules, induction force between polar and nonpolar molecules, and dispersion force among nonpolar molecules. The dynamic contact model of the sphere-sample is analyzed based on the Hamaker hypothesis using the continuum medium method. Assuming that the sample surface is an infinite plane, the van der Waals force (\(F_{\text{vdw}}\)) between the microsphere and the sample surface is calculated as follows [37]:  
\[F_{\text{vdw}}=\frac{TR}{180D^{8}}-\frac{HR}{6D^{2}} \tag{5}\]  
where \(R\) is the microsphere radius, \(D\) is the distance between the microsphere and the sample surface; \(T\) is the expression of \(\pi^{2}\rho_{1}\rho_{2}A\); H is the expression of \(\pi^{2}\rho_{1}\rho_{2}B\), where \(\rho_{1}\) and \(\rho_{2}\) are the densities of the microsphere and the sample surface, respectively; \(A\) is the repulsive force constant; and \(B\) is the attractive force constant. When the absolute distance (\(D\)) between the microsphere and the sample surface is considered as a variable, the change trend of the van der Waals force in the microcontact areas are obtained through simulation. Let \(R=80\) um, and the simulation of the van der Waals force is obtained as shown in Figure 3. Here, the horizontal axis represents the distance between microsphere tip and sample surface, and the longitudinal axis represents the magnitude of van der Waals force. The van der Waals force in section A behaves as a repulsive force. By contrast, the van der Waals force in section B behaves as an attractive force. In other words, when the microsphere approaches the sample surface, the van der Waals force changes the \"needle point force field\" from a long-range attractive force to a short-range repulsive force in the microcontact regions. Capillary force (\(F_{\text{cap}}\)) is the surface tension caused by the wetting action of the water film between sample surface and microsphere. The water film is generated from the air humidity and hydrophilicity of the sample surface [38]. Figure 4 shows the diagram of the capillary force existing in nanocontact areas between microsphere and sample surface, where \(R\) is the radius of the microsphere, \(D\) is the distance between the microsphere and the sample surface, \(r\) is the curvature radius of the water film,  
Figure 3: Simulation of the van der Waals force. \(\theta_{1}\) is the contact angle between the microsphere and the water film, and \(\theta_{2}\) is the contact angle between the sample surface and the water film. When \(R\gg D\), the following is obtained:  
\[F_{\text{cap}}^{R\gg D}=2\pi R\gamma_{\text{Lg}}(\cos(\theta_{1}+\beta)+\cos \theta_{2}-\frac{D}{r})\approx 4\pi R\gamma_{\text{Lg}} \tag{6}\]  
Under the micro-angle condition, the capillary force and the interfacial friction in the modified model (shown in Equation 3) have the same order of magnitude, which means that under certain air humidity, the capillary force may multiply the resultant force of the system (items on the right side of Equations (1) and (2), except \(F_{\text{drive}}\)). This finding proves that, under the atmospheric environment, the influence of capillary force on the dynamic contact model of this probe system cannot be ignored. As main microforces in the mesoscopic scale, the co-action of the van der Waals and capillary forces determines the dynamic mechanical properties of nanocontact between the microsphere tip and the sample surface.