# Nanoparticles Synthesis in Wet-Operating Stirred Media: Investigation on the Grinding Efficiency

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Experimental Set-Up

#### 2.2. Numerical Set-Up

^{−7}s, much smaller than the actual collision time [20,21], ensures simulation stability and full resolution of the particle contacts. Due to the aforementioned initialization procedure, the simulations are run for 0.5 s to let the system settle and reach a pseudo steady-state condition. In the following 0.5 s, the new impacts between the particles and detachments are monitored at every time step, allowing the determination of collision frequency and duration.

## 3. Results

#### 3.1. Experimental Etching Efficiency

#### 3.2. Numerical Characterization of Collisions

#### 3.3. Influence of the Stirring Bar Inclination

#### 3.4. Influence of the Bead Size

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) Detail of the experimental setup. (

**b**) Snapshot of the bottom portion of the simulation domain, with stirring bar and the beads (yellow for ZrO${}_{2}$ and light blue for Ag), partially lifted by the bar motion.

**Figure 2.**Experimental time evolution of the total etched material from the Ag beads at different stirring velocities. The initial total mass of the four Ag beads is about 593 mg.

**Figure 3.**Total (

**a**) translational and (

**b**) rotational kinetic energy for the reference system at different stirring velocities in a time window of $0.5$ $\mathrm{s}$ after the initialization transient.

**Figure 4.**(

**a**,

**d**) Relative kinetic energy, (

**b**,

**e**) impact angle, and (

**c**,

**f**) collision frequency as a function of the stirring velocity for the impacts between all the particles (left column) and those involving at least an Ag particle (right column). In the first and second row the median, first, and third quartiles of the distribution are shown.

**Figure 5.**Map and contours of the joint probability density function for impact angle and relative kinetic energy at (

**a**,

**d**) 300 rpm, (

**b**,

**e**) 600 rpm, and (

**c**,

**f**) 900 rpm for the impacts between all the particles (left column) and those involving at least an Ag particle (right column). On the edges of every map the marginal probability density functions and the corresponding quartiles are also shown (the dashed red lines representing the first and third quartiles and the dashed black line the median).

**Figure 6.**Map and contours of the joint probability density function for impact angle and relative kinetic energy for the impacts between all the particles at 600 $\mathrm{rpm}$ and with stirring bar angle of (

**a**) 30° and (

**b**) 50°. On the edges of every map the marginal probability density functions and the corresponding quartiles are also shown (the dashed red lines representing the first and third quartiles and the dashed black line the median).

**Figure 7.**Map and contours of the joint probability density function for impact angle and relative kinetic energy at (

**a**,

**d**) 300 rpm, (

**b**,

**e**) 600 rpm, and (

**c**,

**f**) 900 rpm for the impacts between all the particles in the bi-disperse (left column) and tri-disperse system (right column). On the edges of every map the marginal probability density functions and the corresponding quartiles are also shown (the dashed red lines representing the first and third quartiles and the dashed black line the median).

**Figure 8.**Collision frequency as a function of the stirring velocity for the impacts between all the particles (

**a**) and those involving at least an Ag particle (

**b**).

**Table 1.**Equations for the Discrete Element Method (DEM) model. Notice that for particle-wall interactions the equivalent radius and mass are those of the particle.

Parameter | Equation |
---|---|

Equivalent radius | ${r}_{\mathrm{r}}={\left(\frac{1}{{r}_{i}}+\frac{1}{{r}_{j}}\right)}^{-1}$ |

Equivalent mass | ${m}_{\mathrm{r}}={\left(\frac{1}{{m}_{i}}+\frac{1}{{m}_{j}}\right)}^{-1}$ |

Equivalent Young modulus | ${E}_{\mathrm{r}}={\left(\frac{1-{\nu}_{i}^{2}}{{E}_{i}}+\frac{1-{\nu}_{j}^{2}}{{E}_{j}}\right)}^{-1}$ |

Equivalent shear modulus | ${G}_{\mathrm{r}}={\left(\frac{2(1+{\nu}_{i})(2-{\nu}_{i})}{{E}_{i}}+\frac{2(1+{\nu}_{j})(2-{\nu}_{j})}{{E}_{j}}\right)}^{-1}$ |

Normal overlap | ${\delta}_{\mathrm{n}}={r}_{i}+{r}_{j}-\u2225{\mathit{x}}_{i}-{\mathit{x}}_{j}\u2225$ |

Tangential overlap | ${\delta}_{\mathrm{t}}={\int}_{{t}_{0}}^{t}{\mathit{v}}_{\mathrm{c},\mathrm{t}}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\mathit{t}\phantom{\rule{0.166667em}{0ex}}d\tau $ |

Normal stiffness | ${k}_{\mathrm{n}}=\frac{4}{3}{E}_{\mathrm{r}}\sqrt{{r}_{\mathrm{r}}{\delta}_{\mathrm{n}}}$ |

Tangential stiffness | ${k}_{\mathrm{t}}=8{G}_{\mathrm{r}}\sqrt{{r}_{\mathrm{r}}{\delta}_{\mathrm{n}}}$ |

Normal damping coefficient | ${\gamma}_{\mathrm{n}}=-2\frac{ln\u03f5}{\sqrt{ln{\u03f5}^{2}+{\pi}^{2}}}\sqrt{\frac{5}{3}{m}_{\mathrm{r}}{E}_{\mathrm{r}}\sqrt{{r}_{\mathrm{r}}{\delta}_{\mathrm{n}}}}$ |

Tangential damping coefficient | ${\gamma}_{\mathrm{t}}=-4\frac{ln\u03f5}{\sqrt{ln{\u03f5}^{2}+{\pi}^{2}}}\sqrt{\frac{5}{3}{m}_{\mathrm{r}}{G}_{\mathrm{r}}\sqrt{{r}_{\mathrm{r}}{\delta}_{\mathrm{n}}}}$ |

**Table 2.**Model parameters and material properties (adapted from [21]).

Zirconia, ZrO${}_{2}$ | Silver, Ag | Glass | PTFE | |
---|---|---|---|---|

Density [$\mathrm{k}\mathrm{g}/{\mathrm{m}}^{3}$] | 6067 | 10490 | 2510 | 2200 |

Young’s Modulus [GPa] | 210.0 | 82.0 | 70.0 | 0.50 |

Poisson ratio | 0.31 | 0.36 | 0.24 | 0.46 |

Coefficient of restitution | 0.92 | 0.80 | 0.99 | 0.80 |

Coefficient of friction | 0.15 | 0.55 | 0.27 | 0.08 |

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**MDPI and ACS Style**

Trofa, M.; D’Avino, G.; Fabiano, B.; Vocciante, M.
Nanoparticles Synthesis in Wet-Operating Stirred Media: Investigation on the Grinding Efficiency. *Materials* **2020**, *13*, 4281.
https://doi.org/10.3390/ma13194281

**AMA Style**

Trofa M, D’Avino G, Fabiano B, Vocciante M.
Nanoparticles Synthesis in Wet-Operating Stirred Media: Investigation on the Grinding Efficiency. *Materials*. 2020; 13(19):4281.
https://doi.org/10.3390/ma13194281

**Chicago/Turabian Style**

Trofa, Marco, Gaetano D’Avino, Bruno Fabiano, and Marco Vocciante.
2020. "Nanoparticles Synthesis in Wet-Operating Stirred Media: Investigation on the Grinding Efficiency" *Materials* 13, no. 19: 4281.
https://doi.org/10.3390/ma13194281