## 2 Material and Methods
### DEM and Simulation Conditions  
The DEM soft-sphere approach originally developed by [PERSON] and [PERSON] (1979) [41] and implemented in the DEM open source code LIGGGHTS version 3.8.0 (DSC, Linz, Austria) was used to model the media ball motions within the attritor mill HD01 (Union Process, Akron, Ohio, USA). The motion of individual particles is determined by solving [PERSON]'s equations, Equations (1) and (2), considering the effect of normal and tangential forces, gravity and torque for a particle i interacting with a particle j:  
\[m_{i}\frac{dv_{i}}{dt}=\sum\left(F_{ij}^{n}+F_{ij}^{t}\right)+m_{i}g \tag{1}\]  
\[I_{i}\frac{d\omega_{i}}{dt}=\sum\left(R_{i}\times F_{ij}^{t}-\tau_{ij}^{r}\right) \tag{2}\]  
where \(F_{ij}^{n}\) and \(F_{ij}^{t}\) are the normal and the tangential forces according to the Hertz-Mindlin (no-slip) contact model, whereas \(m_{i}\), \(I_{i}\), \(v_{i}\), \(\omega_{i}\) and \(\tau_{ij}^{r}\) are the mass, the moment of inertia, the translational velocity, the rotational velocity and the rolling torque of particle \(i\), respectively. \(R_{i}\) is the vector between the centre of particle \(i\) and the contact point where the force \(F_{ij}^{t}\) is applied. The DEM input parameters used for the simulations are listed in the Table 3. As the impeller, the grinding media (5 mm in diameter) and the grinding chamber were all made of yttria-stabilised zirconia the same mechanical properties were utilised for all of them. A Young's modulus considerably lower than the real value was used to significantly reduce the computational time. This is a well-established practise in the literature [22; 42; 43; 44; 45; 46] where values as low as \(10^{6}\) Pa are used, rather than realistic values in the order of \(10^{10}\)-\(10^{11}\) Pa, with little effect on the simulation results. Since, a wide range of velocities exists in the attritor an average value of the coefficient of restitution equal to 0.7 was selected. This fairly high value is justified assuming that the collisions between the grinding media are not expected to be very dissipative in absence of powder. The same value for the coefficient of restitution is used for both media-media and media-wall contacts. The static friction was considered as a parameter to adjust against PEPT data to obtain the correct representation of the media flow field. Three values of static friction coefficient were used (0.15, 0.3 and 0.5) and the same coefficient was used for both media-media and media-wall contacts. The rolling friction coefficient of the grinding media was set equal to zero. The latter parameter always acts to oppose rolling, however, its effect can be neglected when the motion of spherical particles, such as the grinding media, is considered [3; 22; 45; 46; 47]. During the PEPT experiments the tracer location was measured on average every 30 milliseconds (approximately 35 measurements per second) therefore, the same characteristic time was used in the DEM simulations to sample the particle data. DEM Data Postprocessing Using the Coarse Graining Method  
The commercial coarse-graining (CG) software Iota@(Particle-Analytics, Edinburgh, UK) was used for the visualisation of DEM results and the calculation of continuum fields of media properties such as solid fraction and velocity, by averaging the discrete DEM information calculated over the entire simulation time. This software has already been used by the author of this paper to comprehensively investigate the grinding media dynamics inside a vertical stirred mill [24]. The main advantage of the