# Enhanced Modeling for Analysis of Fine Particulate Interactions with Coated Surfaces
## 2 Theoretical Model
The formulae for these three variables above are shown in Formulas (5)-(7):  
\[E^{*}=\frac{1}{\frac{1-v_{1}^{2}}{E_{1}}+\frac{1-v_{2}^{2}}{E_{2}}}, \tag{5}\]  
\[R^{*}=\frac{1}{\frac{1}{R_{1}}+\frac{1}{R_{2}}}, \tag{6}\]  
\[\Gamma=\gamma_{1}+\gamma_{2}-\gamma_{1,2}, \tag{7}\]  
where \(E_{i}(i=1,2)\) and \(v_{i}(i=1,2)\) are the [PERSON]'s modulus and the Poisson's ratio for the two spheres, respectively, \(R_{i}(i=1,2)\) are the radii of the two spheres, \(\gamma_{i}(i=1,2)\) are the surface energies of the two solids, and the \(\gamma_{1,2}\) is the interface energy. The normal force is obtained from Equations (3) and (4):  
\[F_{tt}=\frac{4E^{*}a^{3}}{3R^{*}}-\left(8\pi\Gamma E^{*}a^{3}\right)^{\frac{1} {2}}, \tag{8}\]  
The corresponding relative approach is given by:  
Figure 1.0  
Equations (8) and (9) show the revised JKR model used to calculate the normal contact force on a sphere particle considering the van der Walls force. Consider a surface coating with a structure of uniformly distributed cylindrical crystals of equal height [6], as shown in Figure 1.1 the actual contact area between a particle and the coating corresponds to the base areas of these cylinders. For a circular region with a radius of \(R^{c}\), whose area is \(A^{c}=\pi R^{c2}\), the surface area of the top of the crystal is described as:  
Figure 1.2  
where \(r^{cr}\) is the radius of the base of the cylinder, and \(n\) is the number of the crystals in this region. The aim of using this particular coating structure is to minimize the actual contact area between the particles and the coating surface, thus reducing the influence of van der Waals forces. When a spherical particle comes into contact with this surface, the actual contact area comprises the combined areas of multiple crystal bottoms, as depicted in  
Figure 1.3 Schematic diagram of coated surface. Figure 1.4  
Analogous to Equation (4), the two terms in Equation (11) denote pressure induced by deformation and the van der Waals force, respectively. However, it is crucial to note that the work of adhesion \(\Gamma^{c}\) in Equation (11) does not equate to \(\Gamma\) in Equation (4), as the actual contact area diminishes the adhesive effect. Define a parameter of equivalent area ratio for the coating to describe the ratio of the actual contact area to the projected area of the contact surface:  
Figure 1.5  
Then, Equation (11) can be reformulated as the follow equation:  
Figure 1.6  
Hence, by integrating the pressure over the contact area, we derive the applied normal contact force taking into account the coating:  
Figure 1.7  
Similarly, the corresponding relative approach is:  
Figure 1.8  
Equations (14) and (15) are used to calculate the normal force of a spherical particle when contacting a surface with a coating. This model presupposes that the interaction between the particle and the coating does not alter the deformation characteristics of the contact surface, and that the distribution of pressure resulting from the deformation varies uniformly across the contact surface. The  
Figure 1.9 Schematic diagram of particle contact with coated surface. normal forces derived from the solution of this computational model serve as inputs for subsequent computations. These further calculations determine the damping components, along with forces such as tangential and frictional forces. While performing the DEM simulation, the following iterative calculation steps are adopted:  
1. Compute the positions of the particles and geometry at the current time step;
2. Based on the position data of each unit, determine the relative approach between particle-to-particle or particle-to-geometry interactions;
3. Use the relative approach to calculate the normal force exerted on the particles, utilizing the contact model proposed in this paper;
4. Compute the other forces acting on the particles based on the revised normal force;
5. Update the motion and position of the particles according to these calculated forces;
6. Repeat step one until the iteration process reaches completion.