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Three-Dimensional Two-Temperature Modeling of Ar Loop-Type Inductively Coupled Thermal Plasma for Surface Modification

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In this paper, numerical calculations were made for Ar loop-type inductively coupled thermal plasma (loop-ICTP). The loop-ICTP was developed originally by the authors’ group for rapid surface modification of large areas. Loop-ICTP is sustained with a unique three-dimensional (3D) configuration inside a circular loop quartz tube and on the substrate. A 3D and two-temperature thermofluid thermal plasma model was adopted for this calculation. Mass, momentum, and energy conservation equations were solved using a Maxwell equation for vector potential, an electron energy transport equation, and Saha’s equation in the 3D space. Results indicate that Ar loop-ICTP can be sustained and formed in the loop tube and also on the substrate. Moreover, the heavy particle temperatures reaches 1800–2000 K, whereas the electron temperature is about 10,000 K. Loop size effects on the gas temperature and gas flow field were also investigated using the developed model. Results show that adoption of a larger loop tube can be expected to improve the plasma uniformity on the substrate when applied to rapid surface modification.

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\({\varvec{A}}\) :

Vector potential (Wb/m)

\({\varvec{B}}\) :

Magnetic flux density (T)

\({\varvec{E}}\) :

Electric field strength (V/m)

\(T_{\mathrm{h}}\) :

Heavy particle temperature (K)

\(T_{\mathrm{e}}\) :

Electron temperature (K)

\(\rho\) :

Mass density (kg/m\(^3\))

\({\varvec{u}}\) :

Gas flow velocity (m/s)

p :

Pressure (Pa)

k :

Kinetic energy of turbulence (m\(^2\)/s\(^2\))

\(\epsilon\) :

Turbulence dissipation rate (m\(^2\)/s\(^3\))

\(C_{\mathrm{p}}\) :

Specific heat at constant pressure (J/(kg K))

\(\eta _{\mathrm{heff}}\) :

Effective viscosity for heavy particles (Pa s)

\(\eta _{\mathrm{h}\ell }\) :

Laminar viscosity for heavy particles (Pa s)

\(\eta _{\mathrm{ht}}\) :

Turbulent viscosity for heavy particles (Pa s)

\(\kappa _{\mathrm{heff}}\) :

Effective thermal conductivity for heavy particles (W/(m K))

\(\kappa _{\mathrm{h}\ell }^{\mathrm{tr}}\) :

Translational thermal conductivity for heavy particle in laminar flow (W/(m K))

\(\kappa _{\mathrm{e}\ell }^{\mathrm{tr}}\) :

Translational thermal conductivity for electrons in laminar flow (W/(m K))

\(\kappa _{\mathrm{ht}}\) :

Turbulent thermal conductivity for heavy particles (W/(m K))

\(\mu _{\mathrm{e}}\) :

Electron mobility (m\(^2\)/(V s))

\(\omega\) :

Angular frequency of coil current (rad/s)

\({\varvec{\tau }}\) :

Stress tensor (Pa)

\(P_{\mathrm{in}}\) :

Joule heating power per unit volume (W/m\(^3\))

\(P_{\mathrm{rad}}\) :

Radiation loss power (W/m\(^3\))

\(P_{\mathrm{total}}\) :

Total input power for whole calculation volume (W)

\(\pi {\bar{\Omega }}_{ij}\) :

Collision integral between species \(i-j\) (m\(^2\))

\(Z_j\) :

Internal partition function of species j (–)

\({\bar{v}}_{\mathrm{eAr}}\) :

Relative thermal velocity of electron-Ar (m/s)

\(m_{\mathrm{e}}\), \(m_{\mathrm{Ar}}\) :

Mass of electron and Ar (kg)

\(n_{\mathrm{e}}\), \(n_{\mathrm{Ar}}\), \(n_{{\mathrm{Ar}^+}}\) :

Number density of e, Ar, Ar\(^+\) (m\(^{-3}\))

e :

Electronic charge (C)

\(h_{\mathrm{P}}\) :

Planck’s constant (J s)

\(\mu _0\) :

Permeability for vacuum (H/m)

\(k_{\mathrm{B}}\) :

Boltzmann’s constant (J/K)

\(E_{\mathrm{Ar}}\) :

Ar ionization energy (J)

\(\varepsilon\) :

Dielectric constant (F/m)


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Correspondence to Genki Ozeki or Yasunori Tanaka.

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Ozeki, G., Tanaka, Y., Sugiyama, Y. et al. Three-Dimensional Two-Temperature Modeling of Ar Loop-Type Inductively Coupled Thermal Plasma for Surface Modification. Plasma Chem Plasma Process 41, 85–108 (2021).

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