Transport Properties of Non-Equilibrium Plasmas

Abstract

Simulation of thermal plasma processes requires transport coefficients such as electrical and thermal conductivities, viscosity, and diffusion coefficients. As already shown in chapter “Transport Properties Under Plasma Conditions,” calculation of such coefficients, even under equilibrium conditions (both LTE and LCE), is rather involved. In this chapter, derivations of such coefficients under non-equilibrium conditions will be described, considering:

Nomenclature and Greek Symbols

${\overrightarrow{\mathrm{A}}}_{\upiota}$

Vectorial coefficients for calculating the perturbation function Φ_i

b

Impact parameter defined as the perpendicular distance between the path of a projectile and the center of the field created by the object that the projectile is approaching (m)

b_i

Number of atoms in a molecule of species i

$\overleftrightarrow{{\mathrm{B}}_{\mathrm{i}}}$

Second-order tensor allowing calculation of the perturbation function Φ_i

${\overrightarrow{\mathrm{v}}}_{ \mathrm{c}}$

Center of mass velocity for two particles of type i and j (m/s)

${\overrightarrow{\mathrm{c}}}_{\mathrm{i}}^{\mathrm{j}}$

Vectorial coefficients for calculating the perturbation function Φ_i

${\overrightarrow{\mathrm{d}}}_{\mathrm{i}}$

Diffusion driving force of species i

dA

Elementary surface (m²)

$\mathrm{d} \overrightarrow{\mathrm{r}}$

Elementary volume in ordinary space (m³)

$\mathrm{d} \overrightarrow{\mathrm{v}}$

Elementary volume in velocity space: $\left(\mathrm{d}, \overrightarrow{\mathrm{v}},=,\mathrm{d},{\mathrm{v}}_{\mathrm{x}},, .,, \mathrm{d},{\mathrm{v}}_{\mathrm{y}},, .,, \mathrm{d},{\mathrm{v}}_{\mathrm{z}}\right) \left({\mathrm{m}}^3/{\mathrm{s}}^3\right)$

D_a

Ambipolar diffusion coefficient (m²/s)

D_i

Scalar coefficients for calculating the perturbation function Φ_i

D_ij

Ordinary diffusion coefficient (m²/s)

D ^T_i

Thermal diffusion coefficient (kg/m.s)

D ^θ_ij

Diffusion coefficient reflecting a mass transfer in the mixture, which tends to eliminate the temperature difference between electrons and heavy species

$\overline{{\mathrm{D}}_{\mathrm{AB}}^{\mathrm{X}}}$

Mean value of the diffusion coefficient between species A and B (m²/s)

$\overline{{\mathrm{D}}_{\mathrm{BA}}^{\mathrm{X}}}$

Mean value of the diffusion coefficient between species B and A (m²/s)

$\overline{{\mathrm{D}}_{\mathrm{AB}}^{\mathrm{T}}}$

Mean value of the thermal diffusion coefficient between species A and B (kg/m.s)

$\overline{{\mathrm{D}}_{\mathrm{BA}}^{\mathrm{T}}}$

Mean value of the thermal diffusion coefficient between species B and A (kg/m.s)

${\mathrm{D}}_{\mathrm{i}}^{\uptheta^{*}}$

Non-equilibrium diffusion coefficient

e

Elementary charge (C)

$\overrightarrow{\mathrm{E}}$

Electric field (V/m)

E

Externally applied field (V/m)

f_i’

Distribution function of species i after collisions of the i species

f_i

Distribution function of species i

f ⁰_i

Maxwellian distribution function

${\overrightarrow{\mathrm{F}}}_{\mathrm{i}}$

External force (N)

F_x

Component of the force in the x-direction (N)

g

Relative velocity of the species i and j (m/s)

h

Planck’s constant (h = 6.626 × 10⁻³⁴ J.s)

H_i

Molar enthalpy of species i (kJ/mol)

I ⁰_i

Zero-order approximation of Chapman–Enskog expansion

$\overrightarrow{\mathrm{j}}$

Current density vector

${\overrightarrow{\mathrm{J}}}_{\mathrm{E}}$

Energy flux (W/m²)

${\overrightarrow{\mathrm{J}}}_{\mathrm{n}}$

Particle flux (part/m².s)

${\overrightarrow{\mathrm{J}}}_{\mathrm{px}}$

Momentum flux in x-direction

${\overrightarrow{\mathrm{J}}}_{\mathrm{x}}$

Flux of the quantity x

k

Boltzmann constant (k = 1.38 × 10⁻²³ J/K)

K_i(W _i, θ_ij)

Term taking into account the thermal non-equilibrium when electrons and heavy species collide

ℓ

Mean free path (m)

ℓ _z

Mean free path in z-direction (m)

${\overline{\mathrm{m}}}_{\mathrm{A}}$

Number–density–weighted average mass of the species present in gas A

${\overline{\mathrm{m}}}_{\mathrm{B}}$

Number–density–weighted average mass of the species present in gas B

m_i

Mass of the particle of chemical species i (kg)

M_i

Atomic mass of chemical species i (kg)

n

Number density (m⁻³)

n_i

Number density of chemical species i (m⁻³)

p

Pressure (Pa)

$\overleftrightarrow{\mathrm{p}}$

Pressure tensor (Pa)

$\overrightarrow{\mathrm{q}}$

Heat flux (J/m².s)

${\overrightarrow{\mathrm{q}}}_{\mathrm{i}}$

Flux vector associated with the transport of kinetic energy of particles of species i (J/m².s)

${\overrightarrow{\mathrm{q}}}_{\mathrm{R}}$

Reactional heat flux vector (J/m².s)

${\overrightarrow{\mathrm{q}}}_{\mathrm{z}}$

Heat flux vector in z-direction (J/m².s)

q ^ls_ij

Bracket integral

${\overline{\mathrm{Q}}}_{\mathrm{ij}}^{\mathrm{ls}}$

Product of the reduced collision integral by the cross-sectional area presented by particles considered as hard spheres (m²)

$\overrightarrow{\mathrm{r}}$

Position vector

${\overrightarrow{\mathrm{r}}}^{*}$

Relative position vector $\left({\overrightarrow{\mathrm{r}}}^{*}={\mathrm{r}}_1-{\mathrm{r}}_2\right)$ (m)

r_m

Distance of closest approach (m)

R

Perfect gas constant (R = 8.32 J/K.mole)

$\overleftrightarrow{\mathrm{S}}$

Stress tensor

T

Absolute temperature (K)

T_e

Electron temperature (K)

T_i

Absolute temperature of species i (K)

T_h

Heavy species temperature (K)

T*

Effective collision temperature ${\mathrm{T}}^{*}={\left[\frac{1}{{\mathrm{m}}_{\mathrm{i}}+{\mathrm{m}}_{\mathrm{j}}}\left(\frac{{\mathrm{m}}_{\mathrm{i}}}{{\mathrm{T}}_{\mathrm{i}}}+\frac{{\mathrm{m}}_{\mathrm{j}}}{{\mathrm{T}}_{\mathrm{j}}}\right)\right]}^{-1}$

$\overleftrightarrow{\mathrm{U}}$

Unity tensor

${\overrightarrow{\mathrm{U}}}_{\mathrm{i}}$

Peculiar velocity $\left({\overrightarrow{\mathrm{v}}}_{\mathrm{i}}-{\overrightarrow{\mathrm{v}}}_0={\overrightarrow{\mathrm{U}}}_{\mathrm{i}}\right)$ of particles of species i (m/s)

${\overrightarrow{\mathrm{U}}}_{\mathrm{ix}}$

Peculiar velocity in the x-direction

v_i

Velocity of particle i before collision (m/s)

v ^′_i

Velocity of particle i after collision (m/s)

${\overrightarrow{\mathrm{v}}}_{\mathrm{i}}$

Velocity vector of particle of species i (m/s)

$\overline{\mathrm{v}}$

Mean velocity of particles (m/s)

${\overrightarrow{\mathrm{v}}}_0$

Mass-average velocity (m/s)

V(r)

Interaction potential

${\overrightarrow{\mathrm{V}}}_{\mathrm{ij}}$

Relative velocity before collision $\left({\overrightarrow{\mathrm{V}}}_{\mathrm{i}\mathrm{j}}={\overrightarrow{\mathrm{v}}}_{\mathrm{i}}-{\overrightarrow{\mathrm{v}}}_{\mathrm{j}}\right)$ (m/s)

${\overrightarrow{\mathrm{V}}}_{\mathrm{ij}}^{\prime }$

Relative velocity after collision ${\overrightarrow{\mathrm{V}}}_{\mathrm{i}\mathrm{j}}^{\prime }={\overrightarrow{\mathrm{v}}}_{\mathrm{i}}^{\prime }-{\overrightarrow{\mathrm{v}}}_{\mathrm{j}}^{\prime }$ (m/s)

${\overrightarrow{\mathrm{W}}}_{\mathrm{i}}$

Reduced velocity before collision ${\overrightarrow{\mathrm{W}}}_{\mathrm{i}}={\left(\frac{{\mathrm{m}}_{\mathrm{i}}}{2\mathrm{k}{\mathrm{T}}_{\mathrm{i}}}\right)}^{1/2}{\overrightarrow{\mathrm{V}}}_{\mathrm{i}}$

${\overrightarrow{\mathrm{W}}}_{\mathrm{i}}^{\prime }$

Reduced velocity after collision

x_i

Molar fraction of species i

Z_i

Charge number of the ith species

ΔE_I

Ionization potential (eV)

ε

Incidence azimuthal angle

φ_i or Φ_i

First-order perturbation of the Maxwellian distribution of the Boltzmann equation

Φ

First-order perturbation function of the i^th species with the temperature, T_i

γ²

Reduced energy ${\upgamma}^2=\left({\upmu}_{\mathrm{ij}}/2\mathrm{k}{\mathrm{T}}^{*}\right){\mathrm{g}}^2$

κ

Thermal conductivity (W/m.K)

κ_e

Translational thermal conductivity of electrons (W/m.K)

κ_h

Translational thermal conductivity of heavy species (W/m.K)

κ_r

Reactional thermal conductivity (W/m.K)

κ_int

Internal thermal conductivity (W/m.K)

κ ^′_i

translational thermal conductivity of species i (W/m.K)

μ

Molecular viscosity (kg/m.s)

μ_e

Electron mobility (m²/V.s)

μ_j

Viscosity coefficient of the ith species (kg/m.s)

μ_ij

Reduced mass ${\upmu}_{\mathrm{i}\mathrm{j}}=\frac{{\mathrm{m}}_{\mathrm{i}}{\mathrm{m}}_{\mathrm{j}}}{{\mathrm{m}}_{\mathrm{i}}+{\mathrm{m}}_{\mathrm{j}}}$ (kg)

θ

Ratio of electron to heavy-particle temperatures $\left(\uptheta ={\mathrm{T}}_{\mathrm{e}}/{\mathrm{T}}_{\mathrm{h}}\right)$

θ_ij

Ratio of particle i to particle j temperatures $\left({\uptheta}_{\mathrm{ij}}={\mathrm{T}}_{\mathrm{e}}/{\mathrm{T}}_{\mathrm{h}}\right)$

σ_el

Electrical conductivity (S/m)

σ_ei

Electrical conductivity (S/m)

σ_ii

Differential collision cross section

σ_j,A

elastic collision cross-section (m²)

ν_j,A

elastic collision frequency (s⁻¹)

ξ

order of approximation (m²)

χ

Angle of deflection

Ω

Solid angle

Ω ^(ls)_ij

Collision integral