# Numerical Investigation of a Novel Grinding Device
## 2 Materials and Methods
### Numerical Setup  
The computational domain used to perform the numerical study is schematically reported in Figure 1b. ZrO2 grinding beads are yellow, and Ag precursor beads are blue. The stirring movement is imposed by the counterclockwise rotation around the container axis of the L-bent glass shaft. The motion and collisions of the beads in the grinding system are modeled through the Discrete Element Method [25]. For each particle \(i\), the translational \(\mathbf{u}_{i}\) and angular \(\mathbf{\omega}_{i}\) velocities are computed from [PERSON]'s laws of motion (i.e., the force and torque equations for rigid-body motion):  
\[m_{i}\frac{d\mathbf{u}_{i}}{dt}=\sum_{j}\mathbf{F}_{\mathrm{c}}+m_{i}\mathbf{g}\qquad\qquad l _{i}\frac{d\mathbf{\omega}_{i}}{dt}=\sum_{j}\mathbf{T}_{\mathrm{c}} \tag{1}\]  
Figure 1: (**a**) Schematic representation of the outer-bar wet miller (OWM). (**b**) Simulation domain showing the L-bent rotating glass shaft and the ZrO2 (yellow) and Ag (blue) beads. Adapted from [24]. where \(m_{i}\) and \(I_{i}\) are the particle mass and moment of inertia and \(g\) is the gravity vector. The force \(\mathbf{F}_{\text{c}}\) and the torque \(\mathbf{T}_{\text{c}}\) are due to the \(j\) contacts with other particles or the container and are calculated through the Hertz-Mindlin model [23]:  
\[\mathbf{F}_{\text{c}}=F_{\text{c,n}}\mathbf{n}+F_{\text{c,t}}\mathbf{t}=\left( k_{\text{n}}\delta_{\text{n}}+\gamma_{\text{n}}\frac{d\delta_{\text{n}}}{dt} \right)\mathbf{n}-\left(k_{\text{n}}\delta_{\text{t}}+\gamma_{\text{t}}\frac{d \delta_{\text{t}}}{dt}\right)\mathbf{t} \tag{2}\] \[\mathbf{T}_{\text{c}}=-r_{i}F_{\text{c,t}}\mathbf{n}\times\mathbf{t} \tag{3}\]  
In the previous equations, \(\mathbf{n}\) is the unit normal vector to the contact plane (pointing from \(j\) to \(i\)), and \(\mathbf{t}\) is the direction of tangential motion between the contacting surfaces. The symbols \(\delta_{\text{n}}\) and \(\delta_{\text{t}}\) denote the normal and tangential overlaps, \(k_{\text{n}}\) and \(k_{\text{t}}\) are the normal and tangential stiffnesses, and \(\gamma_{\text{n}}\) and \(\gamma_{\text{t}}\) are the normal and tangential damping coefficients. The tangential component of the contact force is limited by the Coulomb law of sliding friction \(|F_{\text{c,t}}|\leq\mu_{\text{F}}|F_{\text{c,n}}|\), with \(\mu_{\text{F}}\) the sliding friction coefficient. These quantities depend on particle dimensions and material properties such as density, Young's modulus, and Poisson ratio. The values employed in the present study have been taken from the literature and are listed in Table 1. Note that the values of the restitution and friction coefficients are subject to some uncertainty, since they depend on several parameters (e.g., temperature, contact geometry, and applied force) specific to the system they were measured with [26]. In addition, the adopted model assumes a fixed restitution coefficient (independent of the impact velocity) and considers the average value of the friction coefficients for the material pairs. The particle trajectory is finally obtained by integrating the following kinematic equations for position \(\mathbf{x}_{i}\) and orientation \(\mathbf{\theta}_{i}\), starting from the initial values \(\mathbf{u}_{i,0}=\mathbf{\theta}_{i,0}=\mathbf{\omega}_{i,0}=\mathbf{0}\) and a random non-overlapping position inside the domain:  
\[\frac{d\mathbf{x}_{i}}{dt}=\mathbf{u}_{i}\qquad\qquad\frac{d\mathbf{\theta}_{i}}{dt}=\mathbf{ \omega}_{i} \tag{4}\]  
More details about the DEM model used here and the related parameters can be found in [23]. The analysis of the collisions requires examination of the relative velocity between the contacting particle surfaces \(\mathbf{v}_{\text{c}}=\mathbf{v}_{i}-\mathbf{v}_{j}\), where the velocity of the \(i-\)th particle surface \(\mathbf{v}_{i}=\mathbf{u}_{i}+\mathbf{\omega}_{i}\times\mathbf{r}_{i}\) is computed from the quantities in Equation (4) and the position vector from the particle center to the contact point \(\mathbf{r}_{i}\).