COMSOL 6.4 - Hydrogen Boltzmann Analysis

In this model the Boltzmann equation in the two-term approximation is solved for a background gas of molecular and atomic hydrogen. This type of study is normally done to obtain macroscopic transport parameters and source terms that can be used in fluid-type models. Here, the results are only analyzed to test if the computed electron energy distribution function (EEDF), transport parameters, and source terms have reasonable values. The outcome of the analysis allows us to have confidence in the cross-section data.

Model Definition

The Boltzmann equation in the two-term approximation can be written as the divergence of the electron flux in the energy space

with a convection part given by

(1)

and a diffusive part with a diffusive coefficient given by

(2)

Here, F0 is the electron energy distribution function (EEDF) (eV−3/2).

For definitions of the quantities in Equation 1 and Equation 2, see the chapter The Boltzmann Equation, Two-Term Approximation Interface in the Plasma Module User’s Guide. Ref. 1 is a good reference to learn about solving the Boltzmann equation for cold plasma applications.

When the Boltzmann equation has been solved, different macroscopic quantities can be computed by suitable integration of electron impact cross sections over the EEDFs.

Typically, the Boltzmann equation in the two-term approximation is solved in the DC or HF limits. In both these limits, the Boltzmann equation is stationary. In the HF limit it is assumed that the EEDF experiences only the time-averaged electric field.

The DC limit occurs when the collision frequency for energy transfer is much larger than the reduced excitation frequency so that the EEDF can follow the electric field instantaneously. The HF limit occurs when the collision frequency for energy transfer is much smaller than the reduced excitation frequency so that the electrons only experience an averaged electric field.

For the conditions explored in this work we are between the DC and HF limits, and in these cases the time-dependent Boltzmann equation needs to be solved. As shown in this work, this can be done. However, when coupling with a fluid model it is not practical. This topic is discussed in Ref. 2 and the practical approach of using the HF limit is justified by the results that the time-averaged EEDFs are similar to the ones obtained in the HF limit. This approach should be used with caution, since computing macroscopic quantities with a time-averaged EEDF is not necessary equal to time-averaged macroscopic quantities.

The electron collisions are characterized by cross sections that need to be provided by the user. In this model, the background gas is atomic and molecular hydrogen. The electron impact collisions considered, listed in Table 1, are from Ref. 3 and retrieved from Ref. 4.

Table 1: hydrogen electron impact collisions.
Reaction	Formula	Type	Δε (eV)
1	e+H2=>e+H2	Elastic	-
2	e+H2=>e+H2	Vibrational excitation	0.516–1.5
3	e+H2=>e+H2	Excitation	7.93–14.6
4	e+H2=>e+H(1s)+H(2p,2s,3,4,5)	Excitation	14.68–17.53
5	e+H2=>2e+H2+	Ionization	15.4
6	e+H2=>2e+H+H+	Ionization	19
7	e+H2=>2e+H2+	Ionization	15.4
8	e+H2=>2e+H+H+	Ionization	19
9	e+H=>e+H	Elastic	–
10	e+H=>e+H(2p,2s,3,4,5)	Excitation	10.2043–13.0615
11	e+H=>2e+H+	Ionization	13.6057

Results and Discussion

Figure 1 shows an EEDF computed in the HF limit and a time-averaged EEDF together with EEDFs for time instants corresponding to the most depleted and populated tails. The time-dependent EEDF clearly has a low-energy region that changes little in time (close to the HF limit) and a high-energy tail that is modulated in time (close to the DC limit). This is because around 10 eV an important set of electron impact reactions have thresholds, and the frequency for electron energy transfer is much larger than in the low-energy region. This plot clearly shows that for the current operation conditions we are between the DC and HF limits. Figure 1 also shows that the time-averaged EEDF is similar to the one obtained in the HF limit, as discussed in Ref. 2.

Figure 2 through Figure 5 present results for the solution of the Boltzmann equation in the two-term approximation in the HF limit for values of reduced fields between 10 to 500 Td. All results have expected values and trends, which gives us confidence to use this cross-section data in fluid models. To be absolutely sure of the quality of cross-section data, it should be compared with swarm experiments. In practice, this means comparing quantities as electron drift velocity and characteristic energy, as was done by the authors in Ref. 2 and Ref. 4.

Figure 1: Computed EEDF in the HF limit and time averaged. Also shown (dashed) are the EEDFs for time instants corresponding to the most depleted and populated tail.

Figure 2: EEDFs for the for values between 10 and 500 Td in steps of 10 Td.

Figure 3: Mean electron energy as a function of the reduced electric field.

Figure 4: Reduced electron mobility as a function of the mean electron energy.

Figure 5: Ionization rate constants as function of the mean electron energy.

References

1. G.J.M. Hagelaar and L.C. Pitchford, “Solving the Boltzmann Equation to Obtain Electron Transport Coefficients and Rate Coefficients for Fluid Models,” Plasma Sources Sci. Technol., vol. 14, pp. 722–733, 2005.

2. K. Hassouni, A. Gicquel, M. Capitelli, and J. Loureiro, “Chemical Kinetics and Energy Transfer in Moderate Pressure H2 Plasmas Used in Diamond MPACVD Processes,” Plasma Sources Sci. Technol., vol. 8, pp. 494–512, 1999.

3. L. Marques, J. Jolly, and L.L. Alves, “Capacitively Coupled Radio-Frequency Hydrogen Discharges: The Role of Kinetics,” J. Appl. Phys., vol. 102, p. 063305, 2007.

4. IST-Lisbon database, www.lxcat.net, retrieved 2023.

Plasma_Module/Two-Term_Boltzmann_Equation/boltzmann_hydrogen

Modeling Instructions

From the menu, choose .

New

In the window, click

Model Wizard

In the window, click

In the tree, select > .

Click .

Click

In the tree, select > .

Click

Add parameters for the gas number density, reduced angular frequency, and reduced electric field.

Global Definitions

Parameters 1

In the window, under click .

In the window for , locate the section.

In the table, enter the following settings:


Name	Expression	Value	Description
Tgas	300[K]	300 K	Gas temperature
p0	25000[Pa]	25000 Pa	Gas pressure
Ngas	p0/(k_B_const*Tgas)	6.0358E24 1/m³	Gas density
f0	2.45[GHz]	2.45E9 Hz	Excitation frequency
w0	2pif0	1.5394E10 Hz	Angular frequency
wsN	w0/Ngas	2.5504E-15 m³/s	Reduced angular frequency
EN	50[Td]	5E-20 V·m²	Reduced electric field

Solve the Boltzmann equation in the two-term approximation in the HF limit and import electron-electron impact cross sections for H2 and H.

Boltzmann Equation, Two-Term Approximation (be)

In the window, under click .

In the window for , locate the section.

From the list, choose .

Select the checkbox.

In the ω/N text field, type wsN.

In the εmax text field, type 200[V].

Cross Section Import 1

In the toolbar, click

and choose .

In the window for , locate the section.

Click

Browse to the model’s Application Libraries folder and double-click the file H2_marques_xsecs.txt.

Click

Cross Section Import 2

In the toolbar, click

and choose .

In the window for , locate the section.

Click

Browse to the model’s Application Libraries folder and double-click the file H_marques_xsecs.txt.

Click

Set input conditions like the gas temperature and mole fraction for H2 and H.

Boltzmann Model 1

In the window, click .

In the window for , locate the section.

In the Tg text field, type Tgas.

Locate the section. In the table, enter the following settings:


Species	Mole fraction (1)
H2	0.5
H	0.5

Locate the section. Find the subsection. Clear the checkbox.

Clear the checkbox.

Initial Values 1

In the window, click .

In the window for , locate the section.

In the ε0 text field, type 3[V].

In the E/N0 text field, type 35[Td].

The first study solves for two reduced electric fields: One at 50 Td that is used as the initial solution for the time-dependent study, and the other at 35.355 Td, which is the rms value of the first field and is used to compare with the time-averaged EEDF.

HF Limit Stationary

In the window, click .

In the window for , type HF Limit Stationary in the text field.

Locate the section. Clear the checkbox.

Step 1: Reduced Electric Fields

In the window, under click .

In the window for , locate the section.

In the text field, type 35.355[Td] 50[Td].

In the toolbar, click

Create a plot for the EEDF in the HF limit.

Results

EEDF HF Limit vs. Time Averaged

In the toolbar, click

In the window for , type EEDF HF Limit vs. Time Averaged in the text field.

Click to expand the section. From the list, choose .

Line Graph 1

Right-click and choose .

In the window for , locate the section.

From the list, choose .

In the list box, select .

Locate the section. From the list, choose .

Locate the section. In the text field, type be.f.

Locate the section. From the list, choose .

In the text field, type xe_be*root.comp1.be.emax.

Click to expand the section. Select the checkbox.

From the list, choose .

In the table, enter the following settings:


Legends
HF Limit

Click to expand the section. From the list, choose .

EEDF HF Limit vs. Time Averaged

In the window, click .

In the window for , locate the section.

Select the checkbox. In the associated text field, type Energy (eV).

Select the checkbox. In the associated text field, type EEDF (eV-3/2).

Locate the section. Select the checkbox.

Select the checkbox.

In the text field, type 0.

In the text field, type 20.

In the text field, type 1e-7.

In the text field, type 1.

In the toolbar, click

The next step is to solve for the temporal evolution of the EEDF. Add a Time Dependent study and choose to solve the time-dependent EEDF.

Add Study

In the toolbar, click

to open the window.

Go to the window.

Find the subsection. In the tree, select > .

Click the button in the window toolbar.

In the toolbar, click

to close the window.

Study 2

Step 1: Time Dependent

In the window for , locate the section.

In the text field, type range(0,(5/f0-0)/4,5/f0) range(5/f0,(6/f0-(5/f0))/24,6/f0).

Click to expand the section. Find the subsection. From the list, choose .

From the list, choose .

In the list box, select .

In the window, click .

In the window for , locate the section.

Clear the checkbox.

Boltzmann Equation, Two-Term Approximation (be)

In the window, under click .

In the window for , locate the section.

From the list, choose .

Boltzmann Model 1

In the window, under > click .

In the window for , locate the section.

In the pA text field, type p0.

In the E/N text field, type EN*cos(2*pi*f0*t).

Plot the EEDFs at the instants where the tail is at its maximum and minimum values. Also, plot the time-averaged EEDF.

Time Dependent

In the toolbar, click

In the window, click .

In the window for , type Time Dependent in the text field.

Results

Line Graph 2

In the window, right-click and choose .

In the window for , locate the section.

From the list, choose .

In the list, choose and .

Locate the section. From the list, choose .

Locate the section. In the text field, type be.f.

Locate the section. From the list, choose .

In the text field, type xe_be*root.comp1.be.emax.

Locate the section. Find the subsection. From the list, choose .

From the list, choose .

In the toolbar, click

Line Graph 3

Right-click and choose .

In the window for , locate the section.

From the list, choose .

Locate the section. From the list, choose .

Locate the section. In the text field, type timeavg(5/f0, 6/f0, be.f).

Locate the section. From the list, choose .

In the text field, type xe_be*root.comp1.be.emax.

Locate the section. From the list, choose .

Locate the section. Select the checkbox.

From the list, choose .

In the table, enter the following settings:


Legends
Time averaged

In the toolbar, click

Add one last study to parameterize over the reduced field and obtain transport parameters and rate coefficients as functions of the electron mean energy.

Boltzmann Equation, Two-Term Approximation (be)

In the window, under click .

In the window for , locate the section.

From the list, choose .

Add Study

In the toolbar, click

to open the window.

Go to the window.

Find the subsection. In the tree, select > .

Click the button in the window toolbar.

In the toolbar, click

to close the window.

Study 3

Step 1: Reduced Electric Fields

In the window for , locate the section.

In the text field, type range(10,10,500)[Td].

In the window, click .

In the window for , locate the section.

Clear the checkbox.

In the text field, type HF Limit parameterization.

In the toolbar, click

Results

EEDFs

In the toolbar, click

In the window for , type EEDFs in the text field.

Locate the section. From the list, choose .

Line Graph 1

Right-click and choose .

In the window for , locate the section.

From the list, choose .

Locate the section. In the text field, type be.f.

Locate the section. From the list, choose .

In the text field, type xe_be*root.comp1.be.emax.

EEDFs

In the window, click .

In the window for , locate the section.

From the list, choose .

Locate the section.

Select the checkbox. In the associated text field, type Energy (eV).

Select the checkbox. In the associated text field, type EEDF (eV-3/2).

Locate the section. Select the checkbox.

Select the checkbox.

In the text field, type 0.

In the text field, type 200.

In the text field, type 1e-7.

In the text field, type 2.

Locate the section. Clear the checkbox.

In the toolbar, click

Electron Mean Energy vs. E/N

In the toolbar, click

In the window for , type Electron Mean Energy vs. E/N in the text field.

Locate the section. From the list, choose .

Global 1

Right-click and choose .

In the window for , locate the section.

In the table, enter the following settings:


Expression	Unit	Description
be.ebar	V	Mean electron energy

Locate the section. From the list, choose .

Electron Mean Energy vs. E/N

In the window, click .

In the window for , locate the section.

From the list, choose .

Locate the section.

Select the checkbox. In the associated text field, type E/N (Td).

Select the checkbox. In the associated text field, type Mean electron energy (eV).

In the toolbar, click

Locate the section. Clear the checkbox.

Electron Mobility vs. Electron Mean Energy

In the toolbar, click

In the window for , type Electron Mobility vs. Electron Mean Energy in the text field.

Locate the section. From the list, choose .

Global 1

Right-click and choose .

In the window for , locate the section.

In the table, enter the following settings:


Expression	Unit	Description
be.muN	1/(Vms)	Reduced electron mobility

Locate the section. From the list, choose .

In the text field, type be.ebar.

Electron Mobility vs. Electron Mean Energy

In the window, click .

In the window for , locate the section.

Select the checkbox. In the associated text field, type Mean electron energy (eV).

Locate the section. Select the checkbox.

In the text field, type 0.

In the text field, type 19.

In the text field, type 0.

In the text field, type 2e24.

Locate the section. Clear the checkbox.

In the toolbar, click

Ionization Rate Constants vs. Electron Mean Energy

In the toolbar, click

In the window for , type Ionization Rate Constants vs. Electron Mean Energy in the text field.

Locate the section. From the list, choose .

Global 1

Right-click and choose .

In the window for , locate the section.

In the table, enter the following settings:


Expression	Unit	Description
be.k_21	m^3/s	Forward rate constant 21: e+H2=>2e+H2+
be.k_22	m^3/s	Forward rate constant 22: e+H2=>2e+H+H+
be.k_29	m^3/s	Forward rate constant 29: e+H=>2e+H+

Locate the section. From the list, choose .

In the text field, type be.ebar.

Click to expand the section. In the toolbar, click

Ionization Rate Constants vs. Electron Mean Energy

In the window, click .

In the window for , locate the section.

Select the checkbox. In the associated text field, type Mean electron energy (eV).

Select the checkbox. In the associated text field, type Rate constant (m3/s).

Locate the section. Select the checkbox.

Locate the section. From the list, choose .