# Numerical Investigation of a Novel Grinding Device
## 3 Results
### General Characterization of Collisions  
The milling process involves thousands of collisions per second, with a variety of impact dynamics. To better understand the general behavior of the system, it is useful to look at the descriptors of the distributions that characterize the variables of interest. In particular, considering particle-particle collisions, the quartiles of the distribution of impact kinetic energy \(ke_{\text{r}}\) and impact angle \(\theta_{\text{impact}}\) are shown in the first two rows of Figure 3, whereas the last row contains the average collision frequency \(v_{\text{c}}\). While the left-hand column (panels a, b, and c) refers to the impacts between all the beads (either Ag or ZrO2), the right-hand column (panels d, e, and f) is for those involving at least one Ag particle. Figure 3: (**a,d**) Impact kinetic energy, (**b,e**) impact angle, and (**c,f**) collision frequency as a function of the stirring velocity. The left column refers to collisions between all particles in the system, while the right column refers to those involving at least one Ag particle. In the first and second row, the median, first, and third quartiles of the distribution are shown. The impact kinetic energy, which spans different orders of magnitude, as one can expect, increases with the stirring velocity. Interestingly, the trend of the distribution median can be fairly described by a power-law with an exponent around 2.5 (i.e., \(k_{\mathrm{e}}\propto\Omega^{2.5}\)). The median of the impact angle, excluding the 100 rpm case, has an almost constant value of 30\({}^{\circ}\), revealing a prevalent tangential component in the collision velocity. The interquartile range is also almost constant and roughly spans 30\({}^{\circ}\). These results are positive for the production of NPs, since ductile materials (such as Ag) have been shown to exhibit maximum abrasion efficiency at low impact angles [33; 34; 35]. At low \(\Omega\), all the particles are confined on the bottom of the vessel and present a quasi-rigid collective motion determined by the impeller. At higher \(\Omega\), the distance among the particles increases, and the collisions become more rare, thus lowering the impact frequency (see Figure 3c). Similar trends can be encountered by looking at the right-hand column of Figure 3, which illustrates the collisions of the Ag particles (i.e., the NP precursors). Despite the greater density of silver with respect to zirconia, the impact kinetic energy is reduced by a factor of around 2 at 100 rpm and 1.2 at 1200 rpm. Therefore, the impact velocity must be lower. Since Ag only accounts for 4 out of the total 44 particles, one would expect a corresponding reduction in the impact frequency, but this is not the case, as only a reduction to 1/5 was found at all stirring speeds. This is probably because each particle has a different collision frequency, with the Ag ones having higher values, and the curve shown in the figure just represents the total. Notwithstanding such a positive aspect, the decreasing trend of \(v_{\mathrm{c}}\) with \(\Omega\) remains. To maximize NP production, it is necessary to balance all the three relevant quantities described above. Notice that, as shown experimentally in the case of IWM [23], there is an optimal operating range for the impact kinetic energy. Indeed, abrasion (i.e., the removal of small surface portions) does not occur below a critical value of \(k_{\mathrm{e}}\), and plastic deformations or the production of macroparticles takes place when this is too high. Therefore, since \(\theta_{\mathrm{impact}}\) is only slightly affected by \(\Omega\) and it is desirable to maximize \(v_{\mathrm{c}}\), the optimal stirring speed is the minimum that is sufficiently energetic to produce NPs.