# Enhanced Modeling for Analysis of Fine Particulate Interactions with Coated Surfaces
## 2 Theoretical Model  
This paper primarily investigates fine particles, smaller than 100 microns, adhering to solar panel surfaces. To facilitate a comprehensive analysis of force characteristics related to particulate matter, we used an idealized uniform sphere as a representative physical model. Considering the thin atmospheric conditions on Mars, for fine particles, the Knudsen number--pertaining to the airflow circulating around these particles--generally exceeds 0.01. This suggests that the interaction pattern between the airflow and these microscopic entities is largely discontinuous [22]. Additionally, our study primarily focuses on the adsorption characteristics of these fine particles; thus, interactions between these particles and the surrounding discontinuous airflow are not taken into account. The computation of normal forces for particles relies on the Hertzian contact theory, while tangential force calculations are founded on the Mindlin-Dresiewicz work. Both types of forces incorporate damping components. The tangential friction force adheres to the Coulomb law of friction model, and rolling friction is executed under the context-independent directional constant torque model. These force calculations are all consolidated within the Hertz-Mindlin (no slip) physics module of the EDEM software. For particles with small size, van der Waals forces become significant and they tend to stick to each other. The calculation of the normal force, whether between particle-particle or particle-geometry interactions, differs from that in the Hertz-Mindlin (no slip) physics module. The JKR model is developed for this kind of adhesive and elastic interactions. According to the theory of [PERSON] [23; 24], the normal pressure distribution acting over the small circular contact area (shown in Figure 1) of radius \(a\) is expressed as:  
\[p(r)=p_{0}\bigg{[}1-\Big{(}\frac{r}{a}\Big{)}^{2}\bigg{]}^{\frac{1}{2}}, \tag{1}\]  
where \(r(r\in[0,a))\) is the distance from the point to the center point of the contact area, and \(p_{0}\) is the pressure at the center of the contact area. In Figure 1, the dashed arc contour delineates the shape of the two particles prior to contact, while the solid arc contour depicts their shape following the contact. The maximum radius of the contact surface is denoted as '\(a\)' in Figure 1. The \(\alpha_{1,2}\) in Figure 1 are the deformations of the two spheres in the normal direction due to contact, respectively. Additionally, the relative approach of the centroids of the two spheres in the normal direction is defined by:  
\[\alpha=\alpha_{1}+\alpha_{2}, \tag{2}\]  
The total normal force is the integral of the pressure on the contact area:  
\[F_{n}=\int_{0}^{a}p(r)2\pi rdr, \tag{3}\]  
Considering the van der Waals force, the normal contact pressure distribution is written as [23]:  
\[p(r)=p_{def}(r)-p_{van}(r)=\bigg{(}\frac{2E^{*}a}{\pi R^{*}}\bigg{)}\bigg{[}1- \Big{(}\frac{r}{a}\Big{)}^{2}\bigg{]}^{\frac{1}{2}}-\bigg{(}\frac{2E^{*}\Gamma }{\pi a}\bigg{)}^{\frac{1}{2}}\bigg{[}1-\Big{(}\frac{r}{a}\Big{)}^{2}\bigg{]} ^{-\frac{1}{2}}, \tag{4}\]  
Figure 1: Contact surface between two spheres. where \(E^{*}\) is the relative contact compliance, \(R^{*}\) is the relative curvature of the surface, and \(\Gamma\) is the work of adhesion. The first term of this equation \(p_{def}(r)\) signifies the pressure attributed to deformation, while the second term \(p_{van}(r)\) corresponds to the pressure induced by the van der Waals force.