COMSOL 6.4 - Oxygen Boltzmann Analysis

The Boltzmann equation can be solved to validate sets of electron impact collision cross sections. In fact, sets of collision cross sections are traditionally inferred by solving a two-term approximation to the Boltzmann equation and comparing the results to swarm experiments. This model solves the Boltzmann equation in the two-term approximation and compares the computed drift velocity and the characteristic energy to experimental data.

Model Definition

The Boltzmann equation in the two-term approximation can be written as

where f is the electron energy distribution function (EEDF) (eV−3/2) and

(1)

and

(2)

For definitions of the quantities in the equations Equation 1 and Equation 2, see the chapter The Boltzmann Equation, Two-Term Approximation Interface in the Plasma Module User’s Guide.

At zero energy, the condition that energy flux is zero must hold:

and as ε → ∞, f → 0.

When the Boltzmann equation has been solved the drift velocity and characteristic energy for a given reduced electric field can be compared to experimental results. The drift velocity is defined as

where E/Nn is the reduced electric field and μN is the reduced electron mobility which for a DC electric field is

The characteristic energy is defined as

where D is the electron diffusion coefficient

The experimental data comes from Ref. 2.

The EEDF is defined by how electrons gain energy from the electric field and lose (or gain) their energy in collisions with the background gas. The electron collisions are characterized by cross sections that need to be provided by the user. In this model, the background gas is molecular oxygen and the following electron impact collisions are considered (electron impact cross sections are obtained from Ref. 3):

Table 1: table of collisions.
reaction	formula	type	Δε (eV)
1	e+O2=>e+O2	Momentum	0
2	e+O2=>O+O	Attachment	0
3	e+O2=>e+O2(rot)	Excitation	0.02
4	e+O2=>e+O2(v=1)	Excitation	0.19
5	e+O2=>e+O2(v=1)	Excitation	0.19
6	e+O2=>e+O2(v=2)	Excitation	0.38
7	e+O2=>e+O2(v=2)	Excitation	0.38
8	e+O2=>e+O2(v=3)	Excitation	0.75
9	e+O2=>e+O2(v=3)	Excitation	0.75
10	e+O2=>e+O2(a1d)	Excitation	0.977
11	e+O2(a1d)=>e+O2	Superelastic	-0.977
12	e+O2=>e+O2(b1s)	Excitation	1.627
13	e+O2(b1s)=>e+O2	Superelastic	-1.627
14	e+O2=>e+O2(45)	Excitation	4.5
15	e+O2(45)=>e+O2	Superelastic	-4.5
16	e+O2=>e+O+O	Dissociation	6.0
17	e+O2=>e+O+O(1D)	Dissociation	8.4
18	e+O2=>e+O+O(1S)	Dissociation	9.97
19	e+O2=>e+O2+	Ionization	12.06

In a superelastic collision, the electrons gain energy from excited species. The mole fraction of each species is given in the table below and is estimated from typical discharge conditions that occur in a drift tube. The degree of ionization is set to 10−6.

Table 2: table of mole fractions of each species.
species	mole fraction
O2	0.99997
O2(a1d)	1.5E-5
O2(b1s)	1E-5
O2(45)	5E-6

Results and Discussion

Figure 1 plots EEDFs resulting from the solution of the Boltzmann equation in the two-term approximation for different values of the mean electron energy. The EEDF with lowest mean electron energy (2 eV blue line) has a very low population of electrons with energies above the ionization threshold. As the mean electron energy increases, the population of electrons with higher energy increases. This makes ionization processes more likely. Electron impact ionization is important as it is usually the primary mechanism for sustaining plasmas. Notice also that the EEDF is not linear on the log scale. This indicates that the EEDF is non-Maxwellian under these conditions.

The reduced electron transport properties are plotted in Figure 2. The transport properties have a much weaker dependence on the EEDF compared to rate or Townsend coefficients. The electron mobility and electron energy mobility decrease as the mean electron energy increases. The electron diffusivity and electron energy diffusivity increase as the mean electron energy increases. If the EEDF was Maxwellian the rate of change of the transport parameters would be such that the following relations would hold

In the case that the EEDF is non-Maxwellian, this relation does not necessarily hold true.

The electron drift velocity and the characteristic energy (ratio of the electron diffusivity and mobility) are macroscopic quantities that can be computed by integrating appropriate cross sections over computed EEDFs and compared directly with measurements from swarm experiments. Since the EEDF is very sensitive to the cross sections the comparison between the measured and computed drift velocity and the characteristic energy can be used to find a coherent set of cross sections.

The simulated and experimental drift velocity and characteristic energy are plotted in Figure 3 and Figure 4. The agreement between the two is good over a wide range of reduced electric fields, indicating that the cross section data is consistent with experimental measurements.

The fraction of power channeled into various reactions is shown in Figure 5. This is important in design of plasma sources because often the desired reactive species is known in advance. The plot gives an indication of what the target mean electron energy should be in order to channel as much of the available power into a specific reaction. Of course, the power channeled into ionization must be high enough to sustain the plasma.

The Townsend coefficients are plotted in Figure 6. The Townsend coefficients offer an alternative way of defining reaction rates. The reaction rate depends on the electron flux rather than the electron density. Townsend coefficients should be used when modeling DC discharges.