# Magnetic Field Generated during Electric Current-Assisted Sintering: From Health and Safety Issues to Lorentz Force Effects

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Magnetic Field Effects in Electromagnetic Processing of Materials (EPM)

## 2. The Combined Experimental/Computational Methodology

**A**,

**J**, ${\mu}_{0}$ and ${\mu}_{r}$ are the magnetic vector potential, current density, permeability of vacuum, and relative permeability of the material.

## 3. Health Effects and Safe Distance for the Operators

## 4. Magnetic Field Distribution during SPS of Different Materials: Electrical Conductivity and Magnetic Permeability Effects

_{12}O

_{19}. Using a different approach, Li et al. [36] applied an external magnetic field using a inductor coil on iron-based powders using a field strength up to 0.5 T AC 50 Hz. The results suggested an enhanced diffusion of the alloying elements resulting from the application of an external magnetic field. The implications of these magnetic fields are not well-investigated, and it is expected that any magnetic-related effect might have a radial dependence. Further work might be needed to understand and exploit these effects.

## 5. Magnetic Field Effects at Interparticle Contacts

^{2}(6 × 10

^{8}A/m

^{2}) under a discharge time of 50 ms. The effect of the neck growth was investigated for contact areas of 154.94 and 754.77 μm

^{2}.

^{2}for iron. Figure 6 shows the contribution of the Z-axis for iron, where the Lorentz force reached the peak value of 321 N/mm

^{3}, far greater than the 9.65 N/mm

^{3}for copper. This level of force is enough to affect the shape of the particle under such high current density.

^{2}discharge for 0.4 s. The two copper balls were bonded together well at the contact position. Surprisingly, there are many protrusions on the surface of the copper ball that might originate by the molten material being ejected due to the Lorentz force. Figure 7d shows a similar observation when using a current of 7.5 kA/cm

^{2}for 0.6 s. Figure 7e,f map the Lorentz force and magnetic distribution, and the peak value was iron at 15.7 N/mm

^{3}, which is sufficient to promote molten metal ejection under a high current density [40].

## 6. Conclusions

^{2}for 0.4 s, where the molten droplet ejection was correlated with the strong repulsive Lorentz force acting on the molten copper. The results suggest that magnetic-related effects cannot be neglected when using ferromagnetic materials, when using extremely large currents. Further investigation should be dedicated to pulsing and frequency-related effects and how to exploit magnetic-related effects.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Assumed spark plasma sintering (SPS) model’s boundary conditions for (

**A**) electric and (

**B**) thermal fields.

**Figure 2.**Simulation of magnetic field distribution (units in Tesla) using an alumina sample with 30 mm diameter, and assuming a current of 3000 A DC and no sintering chamber (i.e., shield). The safety distance to meet the 5 Gauss line is 119 cm.

**Figure 3.**Magnetic field distribution when using a 30 mm sample of alumina and current of 3000 A, adding a 2 cm-thickness stainless-steel shell. The magnetic field strength at the position represented by the green circle within the chamber is 1.94 × 10

^{−3}T, while the magnetic field outside the chamber (red circle) is 1.218 × 10

^{−7}T. The chamber can effectively shield the magnetic field down to a safe level for the operator.

**Figure 4.**Comparison of magnetic field distribution for different materials assuming a voltage of 10 V; (

**a**) the maximum recorded field strength within the graphite sample was 0.053 T, (

**b**) that for iron was 1.4 T, and (

**c**) that for alumina was as low as 9.1 × 10

^{−10}T. Simulations in (

**d**) were performed at a temperature exceeding the Curie point, reaching 0.014 T within the iron sample.

**Figure 5.**Comparison of electro, thermal, and magnetic effects for 100 µm copper sphere discharged under a current density of 60 kA/cm

^{2}, with the first column for small contact area and second column for large contact area (154.94 and 754.77 μm

^{2}, respectively). (

**a**) Current density distribution graph for small contact (peak current density was 7.59 × 10

^{10}A/m

^{2}) and (

**b**) large contact area (peak current density was 2.45 × 10

^{10}A/m

^{2}). The temperature distribution graph under (

**c**) small and (

**d**) large contact area. Magnetic field distribution with arrows indicating the Lorentz force for (

**e**) small contact area (peak magnetic intensity reached 0.13 T) and for (

**f**) large contact area (peak magnetic intensity reached 0.06 T). Z-axis component Lorentz force contribution for (

**g**) small contact area (max of 9.65 N/mm

^{3}) and (

**h**) for large contact area (max of 1.08 N/mm

^{3}).

**Figure 6.**Lorentz force z-axis contribution (

**a**,

**b**) and magnetic field distribution (

**c**,

**d**) in the case of copper (60 kA/cm

^{2}) in the first column and iron in the second column (200 kA/cm

^{2}).

**Figure 7.**(

**a**) Surface morphology of as-received 5 mm copper spheres. (

**b**,

**c**) Overall appearance with optical microscope and SEM after application of a current of 15 kA/cm

^{2}for 0.4 s. (

**d**) Local appearance with optical microscope at a current 7.5 kA/cm

^{2}for 0.6 s. (

**e**,

**f**) The Lorentz force contribution Z-component and the magnetic field strength distribution were simulated using a current of 15 kA/cm

^{2}for 0.4 s, the Lorentz force peak value was 15.7 N/mm

^{3}, and the peak magnetic field strength was 1.16 T.

**Table 1.**Magnetic field effects in electromagnetic processing of materials (EPM). The possible implications on sintering are listed in italics.

Effect Name | Principle and Mathematical Formulation | Typical Use, Implications on Sintering |
---|---|---|

Biot–Savart Law | This formulation resembles well the magnetic-induced effect in using DC currents. The Biot–Savart Law quantifies the magnetic field strength inside and outside the conductor. It refers to an infinitely long wire with radius r [9,22,23].
$$B=\frac{{\mu}_{0}I}{2\pi {r}_{0}}$$
_{0}—The vacuum permeability, with value of 4π × 10^{−7} H/m.r _{0} is the radius of the conductor | It can be used to calculate the magnetic field strength in the vicinity of a conductor. It can accurately identify the health and safety exposure distance [9,22,23]. Health and safety hazard for the operators and magnetization of the electrically conductive sample under an electrical discharge. |

Skin effect | It commonly occurs in high-frequency alternating current, resulting in preferential current flow within the skin depth.
$$\mathsf{\Delta}=\sqrt{\frac{2}{\omega \mu \gamma}}$$
Where $\omega ,\mu $, and $\gamma $ are angular frenquency, permeability, and conductivity. According to Lenz’s law, the induced current is always in the opposite direction to the imposed current, which eventually leads to the current in the conductor tending to approach the surface of the conductor [10,11,12,24]. Eddy currents are loops of electrical current within a conductor resulting from a changing magnetic field [24,25]. | Heat generation localized on the skin depth. This effect was commonly used in induction heating of a workpiece [10,11,13,24]. Preferential heating by skin effect. Possibility to counterbalance radiative heat losses. |

AC resistance change in the vicinity of the Curie temperature | An alternating (i.e., time-variable) current flowing across a ferromagnetic conductor sees an increase in resistivity due to the skin effect. These effects disappear above the Curie temperature [26,27]. | For ferromagnetic materials (i.e., Ni, Fe, Co), the Curie temperature should be considered during heating [26,27]. Precise calibration of the temperatures and rapid drop of electrical resistance at a temperature greater than the Curie point. |

This effect occurs under high-frequency current (100 kHz) and it was typically used to generate overheating by promoting an interaction between magnetic fields. Eddy currents result in preferential current distribution [11,13,24]. | When using high-frequency alternating current for welding, the proximity effect should be considered [11,13]. Not investigated in sintering. It is expected to provide further overheating at interparticle contact points. | |

Pinch effect | Pinch effect refers to the compressive forces acting on the conductive media as a result of the magnetic forces. The conductive media can be in the form of a liquid, a solid, or a plasma. These effects are particularly strong in the presence of large currents as in the case of a capacitor discharge [14,15,28,29]. | The pinch effect induces a shape change of the sample, easing its extraction out of the die [15]. Might not have strong implications in pressure-assisted sintering. Pinch-related pressure is several orders of magnitude lower than the sintering pressure (≈ MPa). |

This phenomenon was often encountered in the welding. The magnetic bias blow is caused by the unbalanced magnetic field around the electrical arc [17,18,19]. | An AC current is less susceptible to arc blowing compared to DC [17,18,19]. Arcing is usually seen as an undesired effect in sintering. | |

Reactance dependence on duty cycle | Ferromagnetic materials significantly increase the reactance, reducing the welding or sintering current [19].
$$Z=R+jX$$
| Ferromagnetic materials increase the reactance, and leads to an increase in impedance, ultimately resulting in a decrease in current [19]. Implications on the material selected as tooling. |

**Table 2.**Safety distance test under DC conditions, for different currents assuming an exposure limit of 5 Gauss.

Standard Commercialized SPS Types | Dimension Components [mm] | Max Pressing Force [kN] | Max Current [A] | Safety Distance without Shield (DC) [cm] | Biot–Savart Distance Calculation [cm] * |
---|---|---|---|---|---|

HP D 2.5 | Ø 30 | 25 | 3000 | 118.9 | 120 |

HP D 10 | Ø 50 | 100 | 5500 | 217.7 | 200 |

HP D 25 | Ø 80 | 250 | 8000 | 314.7 | 320 |

HP D 60 | Ø 120 | 600 | 16,000 | 629.8 | 640 |

HP D 125 | Ø 150 | 1250 | 24,000 | 956.3 | 960 |

HP D 250 | Ø 300 | 2500 | 48,000/24,000 | 1894/1949 | 1920/960 |

**Table 3.**Electrical conductivity and relative magnetic permeability samples considered in the simulations 20 °C.

Electrical Conductivity (S m) | Relative Permeability | |
---|---|---|

Graphite | 8.3 × 10^{4} | 1 |

Iron | 1.12 × 10^{7} | 200,000 |

Alumina | 1 × 10^{−12} | 1 |

Copper | 5.714 × 10^{7} | 1 |

**Table 4.**Representative sintering parameters for electric current-assisted sintering (ECAS) techniques based on high current density.

Inventor, Year, Reference | Pressure, Discharge Time, Voltage, Current Density | Sintered Material |
---|---|---|

Cremer, US1944 | ≈100 MPa, 10 ms, 5–20 V, ≈60 kAcm ^{−2} | Cu, Al, brass |

Parker, US1968 | 10 MPa, 1 ms, >150 kAcm^{−2}2000 Jcm ^{−3} | Ti, Fe |

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**MDPI and ACS Style**

Deng, H.; Dong, J.; Boi, F.; Saunders, T.; Hu, C.; Grasso, S.
Magnetic Field Generated during Electric Current-Assisted Sintering: From Health and Safety Issues to Lorentz Force Effects. *Metals* **2020**, *10*, 1653.
https://doi.org/10.3390/met10121653

**AMA Style**

Deng H, Dong J, Boi F, Saunders T, Hu C, Grasso S.
Magnetic Field Generated during Electric Current-Assisted Sintering: From Health and Safety Issues to Lorentz Force Effects. *Metals*. 2020; 10(12):1653.
https://doi.org/10.3390/met10121653

**Chicago/Turabian Style**

Deng, Huaijiu, Jian Dong, Filippo Boi, Theo Saunders, Chunfeng Hu, and Salvatore Grasso.
2020. "Magnetic Field Generated during Electric Current-Assisted Sintering: From Health and Safety Issues to Lorentz Force Effects" *Metals* 10, no. 12: 1653.
https://doi.org/10.3390/met10121653