## 2 Dynamic Contact Models and Simulation  
### Mechanical Models  
So far, the dynamic characteristics of atomic force microscopy (AFM) have been well investigated, and mechanical models between probe tip and sample surface in the vertical direction have been established and analyzed [26]. Similar to the operating mechanism of AFM, scanning probe microscopy (SPM) based on a quartz tuning fork has been proposed [27; 28]. However, the AFM only detects the microstructure in the vertical direction. Although AFM scans along the lateral direction, it cannot detect the microstructure in that direction. In the present paper, a 3D triggering method based on the characteristics of a resonant device is proposed to detect the sample surface in three dimensions. This method is based on the principle that the vibration parameters of resonant devices are extremely sensitive to external microforces. Different from conventional trigger probes (e.g., mechanical contact modes and optical noncontact modes), the probe in this study makes use of the tapping mode (TM) in contact with the sample surface in the \(Z\) direction, whereas it performs in friction mode (FM) in the \(X\) and \(Y\) directions, as shown in Figures 1a and 2a. The mechanical models of the present study's probe in vertical (\(Z\) direction) and lateral directions (\(X\) and \(Y\) directions) are established and analyzed with reference to the existing AFM mechanical models. At nanoscale, the interaction between the stylus microsphere tip and the sample surface is a complex process. Different interfacial forces, such as capillary, van der Waals, electrostatic, interfacial friction, and hydrogen bonding forces, influence the interaction [29]. In our previous study [30], we established a simple mass-spring contact mechanical model between the microsphere of the probe and the sample surface in three directions, and the microforces mentioned were only analyzed qualitatively and schematically. To complete the study, surface forces (e.g., van der Waals and capillary forces) and their resultant force are analyzed based on this mechanical model and interfacial tribology. The interfacial adhesive force caused by the resultant force and the contact interfacial friction force  
Figure 1: Tapping mode in the \(Z\) direction: (**a**) Contact model between the probe tip and the sample; (**b**) Schematic of the equivalent microkinetic model. Figure 2: Friction mode in the \(Y(X)\) direction: (**a**) Contact model between the probe tip and the sample; (**b**) Schematic of the equivalent microkinetic model. caused by this interfacial adhesive force are deduced. In view of the interfacial tribology, a new modified dynamic contact model is established. The motion equations are established based on the assumption that the center of the microsphere is the centroid (\(m_{x}\) is the effective mass) and the influencing factors of force equilibrium are considered, such as damping force (\(F_{\mathrm{damp}}\)), elastic restoring force (\(F_{\mathrm{restore}}\)), and driving force (\(F_{\mathrm{drive}}\)) of the probe, as well as the effect of microforces, such as van der Waals force (\(F_{\mathrm{vdw}}\)) and capillary force (\(F_{\mathrm{cap}}\)), amongst others. In the \(Z\) direction, the proposed 3D trigger probe makes contact with the sample surface in TM, similar to the conventional AFM model [31]. Figure 1a shows the schematic illustration in the \(Z\) direction. The probe stays still and vibrates along the \(Z\) direction at its resonant frequency, while the sample is driven upward with a constant speed (\(V_{s}\)). Under the TM condition, the microsphere tip maintains in normal contact with the sample surface. However, a small angular variation may exist in the real experiment condition. The existence of the average tip-sample angle may theoretically generate a lateral force. According to previous research [32], when the angle is small, the lateral force may affect the model slightly.