# Numerical Analysis of the Correlation between Arc Plasma Fluctuation and Nanoparticle Growth–Transport under Atmospheric Pressure

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Model Description

_{2}O

_{3}with a 60-degree tip angle. The anode was an iron plate with 40.0 mm diameter and 10.0 mm thickness. Argon shielding gas was injected at 15.0 L/min from an iron nozzle with a 12.2 mm inner diameter, 1.0 mm thickness, and 20.0 mm length. The current and voltage were set as 150.0 A and 10.5 V, respectively. The ground voltage was set at the anode bottom. To collect nanoparticles effectually, this arc plasma discharge part was covered by an iron case with a 100.0 mm diameter, 80.0 mm length, and 40.0 mm outlet diameter. A slow argon flow was also supplied from the top at 0.25 m/s to prevent metal vapor and nanoparticles from going out upward and to collect all the nanoparticles at the bottom outlet. It is noteworthy that this slow flow did not strongly affect the arc plasma. The temperature of this case was fixed at 300.0 K.

_{m}is the net production rate of mass from the molten pool surface to the fluid region, P denotes pressure, η represents viscosity, σ denotes electrical conductivity, E stands for the electric field, B denotes the magnetic flux density, g represents gravitational acceleration, h denotes enthalpy, λ stands for thermal conductivity, C

_{P}is the specific heat at constant pressure, Q

_{rad}represents the radiation loss, Q

_{con}denotes heat generation by condensation, Φ stands for viscous dissipation, S

_{e}is the net production rate of energy around the molten pool surface, and μ

_{0}represents permeability in vacuum. Subscripts z, r, and θ respectively denote the axial, radial, and azimuthal components. Equations (1)–(5) respectively describe conservation of mass, axial momentum, radial momentum, energy, and electric current density. Equation (6) expresses the relation between the electric current and the magnetic field. The momentum exchange at particle generation is negligible. The current balance considering electron emission and ion recombination at the electrode surfaces are treated as described in an earlier report [32].

_{v}is the net production rate of vapor molecules from the molten pool surface to the fluid region. Subscripts p, v, and s respectively denote particle, vapor, and saturated state. Variable f is defined as f = n

_{p}q, where q is the average monomer number in a particle. Equations (7)–(9) respectively describe conservations of vapor molecules, nanoparticles, and total number of monomers in both gas and particle phases at local positions. These equations are written in Eulerian expressions. Therein, it is assumed that the material vapor molecules move with a flow and gradually form nuclei and small nanoparticles that also move with the flow.

_{p}is the diffusion coefficient of particles derived from [27] as

_{B}is Boltzmann’s constant, T represents the temperature, d denotes the diameter, and l is the mean free path. In addition, D

_{v}is the diffusion coefficient of material vapor obtained from the second viscosity approximation in the literature [35]. J is the homogeneous nucleation rate. q

_{c}represents the number of monomers composing a particle in a critical state, as estimated using the modified self-consistent nucleation theory presented by Girshick et al. [36]. Additionally, β

_{0}is a parameter related to the collision frequency given as [33]:

_{v}denotes the latent heat of vaporization. The emission flux of iron vapor molecules from the molten pool surface can be estimated as approximately $\left({P}_{s}-{P}_{v}\right)/\sqrt{2\pi {m}_{v}{k}_{B}T}$ from the kinetic theory [37]. This growth–transport model obtains the spatial distributions of the number density and mean diameter of nanoparticles with a lower computational cost than those of other models [12,13,14,15,16,17,19,20,21,22,23,24,25,26,27].

## 3. Results and Discussion

_{0}denotes a time when the behavior is regarded as quasiperiodic. The present simulation produces a bell-like shape arc plasma with a maximum temperature of approximately 18,000 K, which agrees with the measurement result obtained using spectroscopy [42]. The inner part of the shielding gas flows into the arc plasma, gains thermal energy, flows out of the arc plasma on the base metal surface, and turns downward to the bottom of the domain. The outer part of the shielding gas flows outward, rolls up, merges with the downward flow, and finally flows out of the bottom exit.

_{1}− t

_{0}). Here, τ represents the time lag. The prime mark denotes fluctuation defined as the difference from the time-averaged value. The fixed position (z

_{0}, r

_{0}) = (28 mm, 6 mm) at the arc plasma’s fringe was selected for the temperature fluctuation as anchoring point A, as depicted in Figure 3.

## 4. Conclusions

- ➢
- The core region of the arc plasma is almost stationary, whereas the fringe fluctuates because of fluid dynamic instability between the arc plasma and the shielding gas.
- ➢
- Numerous small particles are generated around the arc plasma’s fringe because of supersaturation. In the downstream region, the vapor molecules decrease by condensation. The nanoparticles decrease by coagulation. These processes are important contributions to particle growth.
- ➢
- The correlation analysis results suggest that the distribution of growing nanoparticles distant from the arc plasma can be controlled via control of the temperature fluctuation at the arc plasma’s fringe.

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Schematic illustration of nanoparticle production by arc plasma. Metal vapor is generated by the high-temperature plasma. Vapor molecules are transported outside the plasma and therein form nanoparticles.

**Figure 2.**Computational domain of an arc plasma system for nanoparticle production and collection. Iron vapor molecules are emitted from a molten pool and transported outside the arc plasma. Therein, the molecules form nanoparticles through growth processes of nucleation, condensation, and coagulation.

**Figure 3.**Instantaneous distributions of temperature. (

**a**) t = t

_{0}; (

**b**) t = t

_{0}+ 5 ms; (

**c**) t = t

_{0}+ 10 ms. The arc plasma has a bell-like shape with a maximum temperature of approximately 18,000 K. A recirculating flow is observed above the anode.

**Figure 4.**Instantaneous distributions of velocity vectors. Vectors slower than 0.1 m/s are blanked out. (

**a**) t = t

_{0}; (

**b**) t = t

_{0}+ 5 ms; (

**c**) t = t

_{0}+ 10 ms. The arc plasma has a maximum speed of approximately 180 m/s. The outer part of the shielding gas forms a recirculating flow above the anode, merges with the downward flow, and finally flows out of the bottom exit.

**Figure 5.**Time evolution of temperatures at two positions. The position below the electrode tip exhibits almost stationary temperature of approximately 18,000 K, whereas position A at the fringe shows fluctuating temperature around 3000 K.

**Figure 6.**Instantaneous distributions of number density of vapor molecules. (

**a**) t = t

_{0}; (

**b**) t = t

_{0}+ 5 ms; (

**c**) t = t

_{0}+ 10 ms. The vapor molecules are emitted from the molten pool surface and are transported in and around the arc plasma. The number of vapor molecules decreases by condensation in the downstream region.

**Figure 7.**Instantaneous distributions of the number density of nanoparticles. (

**a**) t = t

_{0}; (

**b**) t = t

_{0}+ 5 ms; (

**c**) t = t

_{0}+ 10 ms. At the arc plasma fringe, the vapor molecules change their phase to particles through nucleation, condensation, and coagulation. The nanoparticles decrease in the downstream region because of coagulation.

**Figure 8.**Instantaneous distributions of mean diameters of nanoparticles (cutoff for n

_{p}< 10

^{16}m

^{−3}). (

**a**) t = t

_{0}; (

**b**) t = t

_{0}+ 5 ms; (

**c**) t = t

_{0}+ 10 ms. The nanoparticles have larger sizes in downstream regions because of the growth of condensation and coagulation.

**Figure 9.**Cross-correlation coefficients: (

**a**) between A and B; (

**b**) between A and C. Distant position C exhibits stronger correlation than the more proximate position B.

**Figure 10.**Map of maximum cross-correlation coefficient magnitudes. Temperature fluctuation at the arc plasma’s fringe affects the growth and transport of nanoparticles in a far downstream region.

Property | Parameter |
---|---|

Cathode material | Tungsten with 5.0 wt % of La_{2}O_{3} |

Cathode diameter | 3.2 mm |

Cathode length | 25.0 mm |

Cathode tip angle | 60.0 degrees |

Anode material | Iron |

Anode diameter | 40.0 mm |

Anode thickness | 10.0 mm |

Distance between cathode and anode | 5.0 mm |

Shielding gas nozzle material | Iron |

Shielding gas nozzle inner diameter | 12.2 mm |

Shielding gas nozzle length | 20.0 mm |

Shielding gas nozzle thickness | 1.0 mm |

Shielding gas | Argon |

Shielding gas flow rate | 15.0 L/min |

Current | 150.0 A |

Voltage | 10.5 V |

Outer case material | Iron |

Outer case inner diameter | 100.0 mm |

Outer case length | 80.0 mm |

Outer case outlet diameter | 40.0 mm |

Outer case temperature | 300.0 K |

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**MDPI and ACS Style**

Shigeta, M.; Tanaka, M.; Ghedini, E.
Numerical Analysis of the Correlation between Arc Plasma Fluctuation and Nanoparticle Growth–Transport under Atmospheric Pressure. *Nanomaterials* **2019**, *9*, 1736.
https://doi.org/10.3390/nano9121736

**AMA Style**

Shigeta M, Tanaka M, Ghedini E.
Numerical Analysis of the Correlation between Arc Plasma Fluctuation and Nanoparticle Growth–Transport under Atmospheric Pressure. *Nanomaterials*. 2019; 9(12):1736.
https://doi.org/10.3390/nano9121736

**Chicago/Turabian Style**

Shigeta, Masaya, Manabu Tanaka, and Emanuele Ghedini.
2019. "Numerical Analysis of the Correlation between Arc Plasma Fluctuation and Nanoparticle Growth–Transport under Atmospheric Pressure" *Nanomaterials* 9, no. 12: 1736.
https://doi.org/10.3390/nano9121736