COMSOL 6.4 - Dry Air Boltzmann Analysis

In this model the Boltzmann equation in the two-term approximation is solved for a mixture of nitrogen/oxygen at 80/20 (to represent dry air) to obtain the electron energy distribution function (EEDF) and the macroscopic transport parameters and sources terms that can be used in a fluid-type model.

In a cold plasma the EEDF is rarely a Maxwellian and its shape is strongly dependent on the background gas composition. It is recommended to use an analytic EEDF (for example, Maxwellian) when preparing a model in the or interfaces to simplify the work flow. However, after the model is solving well it is recommended to investigate the effects of a more realistic EEDF.

This investigation can be done with the interface. The first part of the study is to solve for the EEDF for the conditions that the reactor is operating and observe how different the transport parameters and source terms are when using an analytic EEDF. The second part is to incorporate the computed data into the fluid model and see if the results of interest change significantly.

The easiest way to incorporate kinetic effects into a fluid model is to directly use the computed EEDFs that were previously computed using the interface. This way, the rate constants of electron impact reactions (defined with electron impact cross sections) and electron transport properties are computed using the given EEDFs. The second way is to give specific data in the form of reaction rates for some reactions or transport parameters. The first approach is definitely the best since all quantities are dependent of only an imported file and given electron impact cross sections. However, there are cases when using the Local Field Approximation (LFA) or Townsend coefficients are needed that the second approach is the only choice.

The EEDF that can be imported into the fluid model can only be parameterized for the electron mean energy. This means that other parameters (for example, ionization degree, mole fractions of excited states, gas temperature, and gas density) that can influence the EEDF need to be estimated to represent the discharge.

The interface also offers the possibility to solve for the Boltzmann equation in the two-term approximation fully coupled with the fluid model equations. With this approach the EEDF is computed for every time step and space position, and the local species densities, electron density, gas density are used. Please note that this approach is very computational demanding and obtaining a solution is very difficult if not impossible.

In Ref. 1 (in the section “Coefficient for fluid equations”) it is made a good discussion on how to use the results of a Boltzmann equation solver to obtain transport and rate coefficients for fluid models.

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)

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

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.

When the Boltzmann equation has been solved different macroscopic quantities can be computed by suitable integration of electron impact cross sections over the EEDFs as show below. In the DC limit the electron drift velocity, mobility, and diffusivity are defined as

and

The rate coefficients are computed from the EEDF by way of the following integral

The mean electron energy is defined by the integral

(3)

Townsend coefficients are computed using

where νk is the frequency associated with the reaction rate kk, and the power absorbed by the electrons from the electric field in the DC limit is given by

The inputs to the Boltzmann equation are the reduced electric field, the gas temperature, the electron density, the ionization degree, the reduced excitation frequency, the different mole fractions of species with which electrons can collide, and the electron impact cross sections. In this example, data for a discharge operating in the DC limit with negligible ionization degree is computed. It is also assumed that only collisions with ground state of molecular oxygen and nitrogen can influence the EEDF significantly. This means it is assumed that collisions with excited states (direct and superelastic) and with atomic elements coming from dissociative reactions do not affect the EEDF. The gas temperature is used to compute the energy that the electrons receive from a collision with the background gas. This contribution is only of interest if the gas energy is close to the electron energy. With this set of approximations only three input parameters are of interest: the reduced electric field, the mole fraction of oxygen, and the mole fraction of nitrogen. The mole fraction for the mixture nitrogen/oxygen is fixed at 80/20, representing a typical dry air composition at sea level. In order to obtain EEDFs to export the input to the study is the electron mean energy and an ODE is automatically introduced to find the reduced electric field.

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 nitrogen. The electron impact collisions considered are obtained from Ref. 2 and Ref. 3 and are listed in Table 1 and Table 2.

Table 1: Oxygen electron impact collisions.
reaction	formula	type	Δε (eV)
1	e+O2=>e+O2	Elastic	0
2	e+O2=>O+O-	Attachment	0
3	e+O2=>e+O2(rot)	Excitation	0.02
4	e+O2=>e+O2(vibrational)	Excitation	0.19 to 0.75
5	e+O2=>e+O2(electronic)	Excitation	0.97 to 1.627
6	e+O2=>e+O2(e4.5)	Excitation	4.5
7	e+O2=>e+O2(e6.5)	Excitation	6.0
8	e+O2=>e+O2(e8.4)	Excitation	8.4
9	e+O2=>e+O2(e9.97)	Excitation	9.97
10	e+O2=>2e+O2+	Ionization	12.06

Table 2: Nitrogen electron impact collisions.
reaction	formula	type	Δε (eV)
1	e+N2=>e+N2	Elastic	0
2	e+N2=>e+N2(rot)	Excitation	0.02
3	e+N2=>e+N2(vibrational)	Excitation	0.29 to 2.35
4	e+N2=>e+N2(electronic)	Excitation	6.17 to 13
5	e+N2=>2e+N2+	Ionization	15.6
6	e+N2=>2e+N2+	Ionization	18.8

Results and Discussion

Figure 1 shows EEDFs for several electron mean energies. It is possible to observe that these EEDFs deviate considerable from a Maxwellian distribution function.

Figure 2 shows the electron mean energy as a function of the reduced electric field. This information (a function that relates the electron mean energy with the reduced electric field) needs to be provided when using the Local Field Approximation (LFA) in the interface. The LFA assumes that the energy gain by the electrons from the electric field is locally lost in collisions with the background gas. It is a function like the one in Figure 2 that gives the model the information of what energy the electrons have provided an electric field computed by Poisson’s equation.

Figure 3, Figure 4, Figure 5, and Figure 6 show different electron transport properties, Townsend coefficients, and rate constants. This information can be used in a plasma fluid model like the one used in the interface. The information needed for the fluid model depends on the type of reactor and the information that is needed to obtain from the model.

For example, to model a corona discharge in dry air at atmospheric pressure where we only want to know the voltage-current characteristic and to have an idea of the plasma density it is not needed to include explicitly all reactions present in the current example in the fluid model. For the fluid model, a set of 4 electron impact reactions for the ionization and attachment described by Townsend coefficients (data in Figure 4) gives a good estimate of electron creation and destruction in the discharge (reactions between heavy species like ion–ion recombination and photoionization are also needed but they will not be described here). To model a corona discharge the LFA is recommended and in this cases the electron mean energy as a function of the reduced field (data in Figure 3) is also needed. Strictly speaking, the Townsend coefficients can be given to the model directly as a function of the reduced electric field by user define functions. However, in the interface the predefined tables for Townsend coefficients and rate constants are always a function of the electron mean energy and in this case the electron mean energy as a function of the reduced field is needed.

If the question changes to “How much ozone is produced?” a more complex chemistry that involves excited states, electron impact dissociation, and atomic oxygen is needed. If the dissociation degree is low it might not be necessary to include it in the Boltzmann equation analysis but it should be investigated if the EEDFs are influenced by the presence of atomic oxygen and nitrogen. To find which reactions are important for ozone production a literature research is needed. But in any case the simple model with 4 electron impact reactions mentioned in the previous paragraph should be used as the starting point on top of which more reactions are added.

In Figure 4 and Figure 6 the 3-body attachment reaction rate and Townsend coefficient are multiplied by the gas density in cm-3 because the cross section is normalized to the gas density in units of cm-3 as mention in the data from Ref. 3.

Figure 7 shows EEDFs parameterized for electron mean energy. It is a 2D map like this that needs to be exported to be used in the plasma interface. The last instructions in the Model Instructions section in this document show how to prepare and export the EEDFs.

Note on units: In the Plasma Module, the electron energy and electron mean energy appear with units of V but internally are treated as eV. As such, in this context, V should be read as eV. In the interface the extra dimension that represents the electron energy and on which the EEDF is solved appears with units of meters but is treated internally as eV.

Figure 1: EEDFs for a mixture of nitrogen/oxygen at 80/20 for several electron mean energies.

Figure 2: Electron mean energy as a function of the reduced electric field.

Figure 3: Electron transport properties as a function of the electron mean energy.

Figure 4: Ionization and attachment Townsend coefficients as a function of the electron mean energy.

Figure 5: Rate coefficients for electron impact reactions with nitrogen.

Figure 6: Rate coefficients for electron impact reactions with oxygen.

Figure 7: EEDFs (logarithm base 10) where the x-axis represents the electron energy and the y-axis represents the electron mean 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. www.lxcat.net

3. Phelps database, www.lxcat.net, retrieved 2022.

Plasma_Module/Two-Term_Boltzmann_Equation/boltzmann_dry_air

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 to represent the model fraction of O2 and N2. A fixed mixture of 80/20 N2/O2 is used in this example but this ratio can be easily parameterized.

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
xN2	0.8	0.8
xO2	1-xN2	0.2

Definitions (comp1)

Variables 1

In the window, under right-click and choose .

Add a variable to compute the gas density at 300 K. This variable is used to compute the Townsend coefficient and rate constants for the 3-body attachment reaction.

Add also a variable to be used for the maximum energy that the electron energy distribution function is solved.

In the window for , locate the section.

In the table, enter the following settings:


Name	Expression	Unit	Description
Ngas	1[atm]/(k_B_const*300[K])	1/m³
Emax	300[V]	V

Choose to solve the Boltzmann equation in the two-term approximation, increase the number of mesh elements along the energy axis, refine the mesh in the low energy region, and set the electron maximum energy.

Boltzmann Equation, Two-Term Approximation (be)

In the window, under click .

In the window for , locate the section.

From the list, choose .

In the N text field, type 300.

In the R text field, type 50.

In the εmax text field, type Emax.

Cross Section Import 1

In the toolbar, click

and choose .

Import electron impact cross sections for nitrogen and oxygen. Use a node to group all features to simplify navigation.

In the window for , locate the section.

Click

Browse to the model’s Application Libraries folder and double-click the file N2_phelps_xsecs_air.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 O2_phelps_xsecs_air.txt.

Click

10: e+N2=>e+N2(v6), 11: e+N2=>e+N2(v8), 12: e+N2=>e+N2(A3), 13: e+N2=>e+N2(A3), 14: e+N2=>e+N2(B3), 15: e+N2=>e+N2(W3), 16: e+N2=>e+N2(A3), 17: e+N2=>e+N2(B3), 18: e+N2=>e+N2(a1), 19: e+N2=>e+N2(a1), 1: e+N2=>e+N2, 20: e+N2=>e+N2(w1), 21: e+N2=>e+N2(c3), 22: e+N2=>e+N2(E3), 23: e+N2=>e+N2(a1), 24: e+N2=>e+N2(Sum), 25: e+N2=>2e+N2+, 26: e+N2=>2e+N2+, 27: e+O2=>O+O-, 28: e+O2=>O+O-, 29: e+O2=>e+O2, 2: e+N2=>e+N2(rot), 30: e+O2=>e+O2(rot), 31: e+O2=>e+O2(v1), 32: e+O2=>e+O2(v1res), 33: e+O2=>e+O2(v2), 34: e+O2=>e+O2(v2res), 35: e+O2=>e+O2(v3), 36: e+O2=>e+O2(v4), 37: e+O2=>e+O2(a1), 38: e+O2=>e+O2(b1), 39: e+O2=>e+O2(e4.5), 3: e+N2=>e+N2(v1), 40: e+O2=>e+O2(e6.0), 41: e+O2=>e+ O2(e8.4), 42: e+O2=>e+O2(e9.97), 43: e+O2=>2e+O2+, 4: e+N2=>e+N2(v1res), 5: e+N2=>e+N2(v2), 6: e+N2=>e+N2(v3), 7: e+N2=>e+N2(v4), 8: e+N2=>e+N2(v5), 9: e+N2=>e+N2(v5), Cross Section Import 1, Cross Section Import 2

In the window, under > , Ctrl-click to select , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , and .

Right-click and choose .

Electron impact reactions

In the window for , type Electron impact reactions in the text field.

In the window, collapse the node.

Set the mole fractions for nitrogen and oxygen. Choose to plot the Townsend coefficients and to not plot rate coefficients.

Boltzmann Model 1

In the window, click .

In the window for , locate the section.

In the table, enter the following settings:


Species	Mole fraction (1)
N2	xN2
O2	xO2

Locate the section. Find the subsection. Select the checkbox.

Clear the checkbox.

Set the electron mean energies to solve for. Use a logarithmic distribution to have a better sampling at low electron energies where variations are faster.

Study 1

Step 1: Mean Energies

In the window, under click .

In the window for , locate the section.

Click

In the dialog, choose from the list.

In the text field, type 0.5.

In the text field, type 20.

In the text field, type 10.

Click .

In the toolbar, click

Results

EEDF (be)

Prepare plots for the electron energy distribution function, mean electron energy as a function of the reduced electric field, Townsend coefficients, and rate constants.

In the window for , locate the section.

Select the checkbox.

In the text field, type 0.1.

In the text field, type 300.

In the text field, type 1e-7.

In the text field, type 10.

Select the checkbox.

Locate the section. From the list, choose .

In the list, choose , , , , and .

In the toolbar, click

Mean Electron Energy (be)

In the window, click .

In the window for , locate the section.

Select the checkbox.

Transport Properties (be)

In the window, click .

In the window for , locate the section.

Select the checkbox.

Locate the section. From the list, choose .

Townsend Coefficients (be)

In the window, click .

In the window for , locate the section.

Select the checkbox.

In the text field, type 0.

In the text field, type 20.

In the text field, type 1e-25.

In the text field, type 2e-19.

Locate the section. From the list, choose .

Global 1

In the window, expand the node, then click .

In the window for , locate the section.

Click

In the table, enter the following settings:


Expression	Unit	Description
be.alpha_25+be.alpha_26	m^2	N2 ionization
be.alpha_43	m^2	O2 ionization
be.alpha_27*Ngas[cm^3]	m^2	3-body attachment
be.alpha_28	m^2	2-body attachment

In the toolbar, click

N2 Rate Coefficients

In the toolbar, click

In the window for , type N2 Rate Coefficients in the text field.

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

In the text area, type N2 Rate Coefficients.

Locate the section. Select the checkbox.

Select the checkbox.

In the text field, type Mean electron energy (V).

In the text field, type Rate coefficient (m<sup>3</sup>/s).

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-20.

In the text field, type 1e-12.

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_1	m^3/s	Elastic
be.k_2	m^3/s	Rotational
be.k_3+be.k_4+be.k_5+be.k_6+be.k_7+be.k_8+be.k_9+be.k_10+be.k_11	m^3/s	Vibrational
be.k_12+be.k_13+be.k_14+be.k_15+be.k_16+be.k_17+be.k_18+be.k_19+be.k_20+be.k_21+be.k_22+be.k_23+be.k_24	m^3/s	Electronic
be.k_25+be.k_26	m^3/s	Ionization

Locate the section. From the list, choose .

In the text field, type be.ebar.

In the toolbar, click

O2 Rate Coefficients 1

In the window, right-click and choose .

In the window for , type O2 Rate Coefficients 1 in the text field.

Locate the section. In the text area, type O2 Rate Coefficients.

Global 1

In the window, expand the node, then click .

In the window for , locate the section.

Click

In the table, enter the following settings:


Expression	Unit	Description
be.k_29	m^3/s	Elastic
be.k_30	m^3/s	Rotational
be.k_31+be.k_32+be.k_33+be.k_34+be.k_35+be.k_36	m^3/s	Vibrational
be.k_37+be.k_38+be.k_39+be.k_40+be.k_41+be.k_42	m^3/s	Electronic
be.k_43	m^3/s	Ionization
be.k_27*Ngas[cm^3]	m^3/s	3-body attachment
be.k_28	m^3/s	2-body attachment

In the toolbar, click

In the following, two datasets are created to prepare the data to plot the EEDFs in a 2D plot and to export the EEDFs to be used in the interface.

The interface is a 0D interface and uses an extra dimension to represent the electron energy on a 1D axis. As the extra dimension is normalized to the value, it has to be manually scaled with the value before exporting the data. To do this, we can use a dataset. With a dataset, you can scale, rotate, and move datasets.

Parametric Extrusion 1D 1

In the toolbar, click