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Hydrogen Global Model Coupled with the Two-Term Boltzmann Equation
Introduction
This tutorial studies the chemistry of a hydrogen plasma in moderate pressures. The main goals are to show how to prepare a global model to study a chemistry and study the influence of the electron energy distribution function (EEDF). This are important steps that need to be done before moving to a space dependent model.
In this model, it is shown how to solve a plasma global model fully self-consistently with the Boltzmann equation in the two-term approximation.
The chemistry and model here presented are based on the work from Ref. 1 and Ref. 2. The results are in general good agreement with the ones from Ref. 1. However, a detailed comparison is not attempted since the models and chemistry have important differences. The reference used for the plasma chemistry and the chemistry here is closer to the simplified chemistry presented in Ref. 2 which does not include negative ions. Regarding the model, the main differences are that the model used here: (i) does not compute the vibrational temperature or solve for the vibrational states, and (ii) solves for the electron mean energy equation which uses power as an input.
Model Definition
The model used in this work considers that the spatial distribution of the different quantities in the plasma reactor can be treated as uniform. Without the spatial derivatives the numerical solution of the equation set becomes considerably simpler and the computation time is reduced. These advantages make a global model a good first approach to study a plasma reactor, especially when complex chemistries are involved or the influence of the EEDF is to be studied.
When using a global plasma model the species densities and the electron temperature are treated as volume-averaged quantities. Detailed information on the global model can be found in the section Theory for Global Models in the Plasma Module User’s Guide. For heavy species the following equation is solved for the mass fraction
where ρ is the mass density (SI unit: kg/m3), wk is the mass fraction, wf,k is the mass fraction in the feed, mf and mo are the mass-flow rates of the total feed and outlet, and Rk is the rate expression (SI unit: kg/(m3·s)). The fourth term on the right-hand side accounts for surface losses and creation, where Al is the surface area, hl is a dimensionless correction term, V is the reactor volume, Mk is the species molar mass (SI unit: kg/mol) and Rsurf,k,l is the surface rate expression (SI unit: mol/(m2·s)) at a surface l. The last term is introduced because the species mass balance equations are written in the nonconservative form and it used the mass-continuity equation to replace for the mass density time derivative. In the last term Mf,l is the inward mass flux of surface l (SI unit: kg/(m2·s)). The sum in the last two terms is over all surfaces where there are surface reactions.
To take possible variations of the system’s total mass or pressure into account, the mass-continuity equation can also be solved:
.
The electron number density is obtained from electroneutrality:
Using the local energy approximation (LEA), the electron energy density nε (SI unit: V/ m3) is computed from
where Rε is the electron energy loss due to inelastic and elastic collisions, Pabs is the power absorbed by the electrons (SI unit: W), and e is the elementary charge. The last term on the right side accounts for the kinetic energy transported to the surface by electrons and ions. The summation is over all positive ions, εe is the mean kinetic energy lost per electron lost, εi is the mean kinetic energy lost per ion lost, and Na is Avogadro’s number. If using the local field approximation (LFA) the electron mean energy equation is not solved and the electron mean energy can be: (i) provided as a function of the electric field; or (ii) obtained by solving the Boltzmann equation in the two-term approximation.
The rate coefficients for electron impact reactions can be computed by appropriate averaging of cross sections over an EEDF. The EEDF can either be analytic or obtained by solving the steady-state Boltzmann equation in the two-term approximation coupled with the equation system (The Boltzmann Equation, Two-Term Approximation Interface in the Plasma Module User’s Guide). When solving for the EEDF, the coupling between the equations is as follows: (i) if the LEA is used, the electron mean energy obtained from the electron mean energy equation is given as input to the Boltzmann solver; (ii) if the LFA is used, the reduced electric field must be given as input to the Boltzmann solver and the electron mean energy comes from averaging over the computed EEDF.
This work uses the LEA. The first part uses a Maxwellian EEDF and the subsequent part solves the Boltzmann equation in the two-term approximation (B2T).
Coupling the B2T with a space-dependent model is computationally very expensive. Therefore, it is recommended to first explore the influence of the EEDF using a global (volume-averaged) model, since it can run simulations in a fraction of the time of a space-dependent model while retaining the tendencies of volume-averaged physical quantities.
The present study also solves the gas heat equation
where Cp is the specific heat at constant pressure of the mixture, T is the gas temperature, hf,k is the enthalpy of species k in the feed, hk is the enthalpy of species k. The heat source (SI unit: W/m3) is given as:
where Δεj is the electron energy loss from reaction j (SI unit: V) and F is the Faraday constant (SI unit: C/mol). The last term in the equation above is only added for electron impact reactions to account for the energy loss or gain by the electron. Note that the electron enthalpy is set to zero and does not contribute to Hj. For electron impact reactions resulting in excitation and ionization Δεj corresponds to the energy of the excited state being excited/deexcited or ionized, for attachment Δεj is set to zero, and for elastic collisions
where me and mk are the electron and heavy species mass in kg, Te is the electron temperature in eV, and Tgas is the gas temperature in K. Heat losses by transport are including in a simplified form
where k is the thermal conductivity of the mixture, TS is the surface temperature, and ΛS is the diffusion length.
Plasma Chemistry
The plasma chemistry is based on the simplified chemistry presented in table 4 of Ref. 2 and is presented in Table 1. The electron impact cross sections are from Ref. 4 and were retrieved from Ref. 5, and the rates are from Ref. 6 and Ref. 7. The model includes eight species: electrons, H2, H, H2+, H+, H3+, and two excited states of hydrogen corresponding to the levels n = 2 and n = 3 that are represented by Hn2 and Hn3.
Table 1: table of collisions and reactions modeled.
Reaction
Formula
Type
1
e+H2=>e+H2
Elastic
-
2
e+H2=>e+H2
Vibrational excitation
0.516-1.5
3
e+H2=>e+2H
Dissociation
7.93-11.72 and17.22-17.53
4
e+H2=>e+H2
Excitation
12.4-14.6
5
e+H2=>e+H+Hn2
Dissociative excitation
14.68
6
e+H2=>e+H+Hn3
Dissociative excitation
16.57
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+Hn2
Excitation
10.2043
11
e+H=>e+Hn3
Charge transfer
12.1
12
e+H=>e+H
Excitation
12.755 and 13.0615
13
e+H=>2e+H+
Ionization
13.6057
14
e+H3+=>3H
Recombination
0
15
e+H2+=>H+Hn2
Recombination
0.01
16
e+H2+=>H+Hn3
Recombination
0.01
17
e+H+=>Hn2
Recombination
0
18
e+H+=>Hn3
Recombination
0
19
Hn2+H2=>H3++e
Ionization
-
20
Hn3+H2=>H3++e
Ionization
-
21
H2+H2+=>H3++H
Ionization
-
22
H2+H2=>2H+H2
Dissociation
-
23
2H+H2=>H2+H2
Association
-
24
H2+H=>3H
Dissociation
-
26
3H=>H2+H
Association
-
In addition to the volume reactions, the surface reaction listed in Table 2 are also implemented.
Table 2: Surface reactions.
Reaction
Formula
Sticking coefficient
1
H=>0.5H2
0.02
2
Hn2=>H
1
3
Hn3=>H
1
4
H+=>H
1
5
H2+=>H2
1
6
H3+=>H2+H
1
Results and Discussion
In Ref. 1, a key parameter in the experiments is the power density that is estimated from the volume of the plasma and the input power. In the global model presented here, the volume needs to be fixed and the parameterization is made over power, which is an input to the electron mean energy equation. The dimensions of the domain were found to match the values of the power density given in table 1 of Ref. 1. These dimensions are used to estimate losses by transport and the plasma volume. However, if the dimensions that are used to estimate the diffusion length are used also to estimate the heat losses by transport the temperature obtained is too low. This makes sense since the region that the plasma occupies is smaller than the reactor chamber Ref. 2. When using dimensions closer to a typical reactor chamber (from Ref. 2) to estimate the heat losses by transport, the gas temperature is in good agreement with Ref. 1.
When making the parameterization in power note that the pressure varies according to table 1 of Ref. 1.
Figure 1 shows the computed EEDFS for the different power densities under study. The EEDFS are similar to the ones obtained in Ref. 1 and present the same behavior that the high-energy tail decreases with increasing power density. Present in Figure 1 is also a Maxwellian EEDF corresponding to the last power density. As can be seen, the computed EEDF deviates strongly from a Maxwellian. However, quantities such as the gas temperature, electron temperature, electron density, H mole fraction, and neutral number densities do not change significantly, as shown in figures Figure 2 through Figure 5.
Figure 2 shows the gas temperature and the H mole fraction as a function of the power density. These results are in good agreement with the results presented in figure 18 of Ref. 1.
The ion number densities presented in Figure 3 also agree well with results in figure 11 of Ref. 1, the dominant ion being H3+ with densities between 1011 and 1012 cm−3 and H2+ being the ion with lowest density.
Figure 6 shows the relative rates of the main ionization mechanisms. The dominating ionization mechanism is the collision quenching of H-excited states that creates H3+. Electron impact ionization of H2 has some importance at low powers but its relative contribution decreases fast with power. Ionization from H atoms plays a minor role. The results are in agreement with those in figure 12(a) of Ref. 1.
Figure 1: Computed EEDF for different power densities and a Maxwellian EEDF corresponding to the highest power density.
Figure 2: Gas temperature and H atom mole fraction as a function of the power density using a Maxwellian EEDF (solid) and solving for the B2T(dashed).
Figure 3: Ion number densities as a function of the power density using a Maxwellian EEDF (solid) and solving for the B2T (dashed).
Figure 4: Electron temperature as a function of the power density using a Maxwellian EEDF (solid) and solving for the B2T(dashed).
Figure 5: Neutrals number density as a function of the power density using a Maxwellian EEDF (solid) and solving for the B2T(dashed).
Figure 6: Relative ionization rates as a function of the power density using a Maxwellian EEDF (solid) and solving for the B2T(dashed).
References
1. 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 Science and Technology, vol. 8, pp. 494–512, 1999.
2. K. Hassouni, F. Silva, A. Gicquel, “Modelling of Diamond Deposition Microwave Cavity Generated Plasmas,” J. Phys. D: Appl. Phys., vol. 43, pp. 153001, 2010.
3. 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 Science and Technology, vol. 14, pp. 722–733, 2005.
4. L. Marques, J. Jolly and L.L. Alves, “Capacitively Coupled Radio-Frequency Hydrogen Discharges: The Role of Kinetics,” J. Appl. Phys., vol. 102, pp. 063305, 2007.
5. IST-Lisbon database, www.lxcat.net, retrieved 2023.
6. M. Capitelli, C.M. Ferreira, B.F. Gordiets and A.I. Osipov, Plasma Kinetics in Atmospheric Gases, Springer, 2000.
7. R.K. Janev, W.D. Langer, K. Evans, Jr. and D.E. Post, Jr. Elementary Processes in Hydrogen-Helium Plasmas, Springer-Verlag, 1987.
Application Library path: Plasma_Module/Global_Modeling/hydrogen_global_model
Modeling Instructions
From the File menu, choose New.
New
In the New window, click  Model Wizard.
Model Wizard
1
In the Model Wizard window, click  2D Axisymmetric.
2
In the Select Physics tree, select Plasma > Plasma (plas).
3
Click Add.
4
Click  Study.
5
In the Select Study tree, select General Studies > Stationary.
6
Click  Done.
Geometry 1
1
In the Model Builder window, under Component 1 (comp1) click Geometry 1.
2
In the Settings window for Geometry, locate the Units section.
3
From the Length unit list, choose cm.
Global Definitions
Parameters 1
Add some parameters to be used in the model.
1
In the Model Builder window, under Global Definitions click Parameters 1.
2
In the Settings window for Parameters, locate the Parameters section.
3
In the table, enter the following settings:
Name
Expression
Value
Description
W
2.5[cm]
0.025 m
Plasma width
H
3[cm]
0.03 m
Plasma height
Wg
5[cm]
0.05 m
Width for heat
Hg
6[cm]
0.06 m
Height for heat
p0
5[kPa]
5000 Pa
Pressure
pw
300[W]
300 W
Power
Qs
500
500
Mass flow
pd
pw/Vol
5.093E6 W/m³
Power density
Vol
(pi*W^2*H)
5.8905E-5 m³
Plasma volume
f0
2.45e9[Hz]
2.45E9 Hz
Excitation frequency
w0
f0*2*pi
1.5394E10 Hz
Angular frequency
hl_ions
0.05
0.05
Correction factor for ions
Definitions
Variables 1
1
In the Model Builder window, under Component 1 (comp1) right-click Definitions and choose Variables.
Define variables for diffusion lengths.
2
In the Settings window for Variables, locate the Variables section.
3
In the table, enter the following settings:
Name
Expression
Unit
Description
Lamb_diff
((pi/H)^2+(2.405/W)^2)^-0.5
m
Diffusion length for species
Lamb_diff_heat
((pi/Hg)^2+(2.405/Wg)^2)^-0.5
m
Diffusion length for heat
Interpolation 1 (int1)
1
In the Definitions toolbar, click  Interpolation.
Define an interpolation function that gives the pressure as a function of power. This function is used to set the pressure of the reactor.
2
In the Settings window for Interpolation, locate the Definition section.
3
In the Function name text field, type pfnc.
4
In the table, enter the following settings:
t
f(t)
300
1400
600
2500
1000
5200
1500
8400
2000
11000
5
Locate the Interpolation and Extrapolation section. From the Extrapolation list, choose Linear.
6
Locate the Units section. In the Function table, enter the following settings:
Function
Unit
pfnc
Pa
7
In the Argument table, enter the following settings:
Argument
Unit
t
W
Set the domain dimensions. The volume and surface areas used in the global model of the reactor are obtained automatically from this geometry.
Geometry 1
Rectangle 1 (r1)
1
In the Geometry toolbar, click  Rectangle.
2
In the Settings window for Rectangle, locate the Size and Shape section.
3
In the Width text field, type W.
4
In the Height text field, type H.
5
Click  Build All Objects.
Plasma (plas)
Choose to solve for a global model of a constant pressure reactor and include the heavy species energy equation.
1
In the Model Builder window, under Component 1 (comp1) click Plasma (plas).
2
In the Settings window for Plasma, locate the Diffusion Model section.
3
From the Diffusion model list, choose Global.
4
Click to expand the Heavy Species Energy Balance section. Select the Include heavy species energy conservation equation checkbox.
5
Locate the Reactor section. From the Reactor type list, choose Constant pressure.
Plasma Model 1
Set the pressure (using the interpolation function previously defined), mass flow, power absorbed by the electrons, an estimation of the plasma sheath voltage drop (for the mean kinetic energy lost per ion lost), and the diffusion length for the heat equation.
1
In the Model Builder window, under Component 1 (comp1) > Plasma (plas) click Plasma Model 1.
2
In the Settings window for Plasma Model, locate the Model Inputs section.
3
In the pA text field, type pfnc(pw).
4
Locate the Total Mass Flow section. In the Qsccm text field, type Qs.
5
Locate the Mean Electron Energy Specification section. In the Pabs text field, type pw.
6
In the εi text field, type 10[V].
7
Locate the Heat Transfer to Surfaces section. In the ΛS text field, type Lamb_diff_heat.
The Plasma Chemistry Import feature
The next steps have instructions to use the Plasma Chemistry Import feature to import a file that automatically creates the hydrogen plasma chemistry.
The following is set or created automatically:
a
Species properties
b
Reaction group features for hydrogen
c
Surface reactions
The documentation accompanying the Plasma Chemistry Import feature contains more information about the file structure and what can be set automatically.
Plasma Chemistry Import 1
1
In the Physics toolbar, click  Global and choose Plasma Chemistry Import.
2
In the Settings window for Plasma Chemistry Import, locate the Plasma Chemistry Import section.
3
Click  Browse.
4
Browse to the model’s Application Libraries folder and double-click the file H2_plasma_chemistry.txt.
5
Click  Import.
Set some properties of the species and the surface reactions.
Set H2 to be the species that the mass fraction is found from mass constraint.
Species: H2
1
In the Model Builder window, expand the Component 1 (comp1) > Plasma (plas) > Group - Species node, then click Species: H2.
2
In the Settings window for Species, locate the Species Formula section.
3
Select the From mass constraint checkbox.
4
Locate the General Parameters section. In the xfeed text field, type 1.
Group - Species
In the Model Builder window, collapse the Component 1 (comp1) > Plasma (plas) > Group - Species node.
Set different aspects of surface reactions: diffusion length, recombination of H at the surface (set in the Forward sticking coefficient field), and add a correction factor for ions (this is an estimate of the drop of the ion density near the surface to better estimate the losses by transport).
1: H=>0.5H2
1
In the Model Builder window, expand the Component 1 (comp1) > Plasma (plas) > Group - Surface Reactions node, then click 1: H=>0.5H2.
2
Select Boundary 2 only.
3
In the Settings window for Surface Reaction, locate the Reaction Formula section.
4
From the Specify reaction using list, choose Sticking coefficient and diffusion.
5
Locate the Reaction Parameters section. In the Λeff text field, type Lamb_diff.
2: Hn2=>H
1
In the Model Builder window, click 2: Hn2=>H.
2
In the Settings window for Surface Reaction, locate the Boundary Selection section.
3
From the Selection list, choose All boundaries.
4
Locate the Reaction Formula section. From the Specify reaction using list, choose Sticking coefficient and diffusion.
5
Locate the Reaction Parameters section. In the Λeff text field, type Lamb_diff.
3: Hn3=>H
1
In the Model Builder window, click 3: Hn3=>H.
2
In the Settings window for Surface Reaction, locate the Boundary Selection section.
3
From the Selection list, choose All boundaries.
4
Locate the Reaction Formula section. From the Specify reaction using list, choose Sticking coefficient and diffusion.
5
Locate the Reaction Parameters section. In the Λeff text field, type Lamb_diff.
4: H+=>H
1
In the Model Builder window, click 4: H+=>H.
2
In the Settings window for Surface Reaction, locate the Boundary Selection section.
3
From the Selection list, choose All boundaries.
4
Locate the Reaction Formula section. From the Specify reaction using list, choose Bohm velocity.
5
Locate the Reaction Parameters section. In the hl text field, type hl_ions.
5: H2+=>H2
1
In the Model Builder window, click 5: H2+=>H2.
2
In the Settings window for Surface Reaction, locate the Boundary Selection section.
3
From the Selection list, choose All boundaries.
4
Locate the Reaction Formula section. From the Specify reaction using list, choose Bohm velocity.
5
Locate the Reaction Parameters section. In the hl text field, type hl_ions.
6: H3+=>H2+H
1
In the Model Builder window, click 6: H3+=>H2+H.
2
In the Settings window for Surface Reaction, locate the Boundary Selection section.
3
From the Selection list, choose All boundaries.
4
Locate the Reaction Formula section. From the Specify reaction using list, choose Bohm velocity.
5
Locate the Reaction Parameters section. In the hl text field, type hl_ions.
Group - Surface Reactions
In the Model Builder window, collapse the Component 1 (comp1) > Plasma (plas) > Group - Surface Reactions node.
Mesh 1
1
In the Model Builder window, under Component 1 (comp1) click Mesh 1.
2
In the Settings window for Mesh, locate the Physics-Controlled Mesh section.
3
From the Element size list, choose Extremely coarse.
In the following prepare a parameterization of the input power from 300 to 2000 W.
Study 1
1
In the Model Builder window, click Study 1.
2
In the Settings window for Study, locate the Study Settings section.
3
Clear the Generate default plots checkbox.
Step 1: Stationary
1
In the Model Builder window, under Study 1 click Step 1: Stationary.
2
In the Settings window for Stationary, click to expand the Study Extensions section.
3
Select the Auxiliary sweep checkbox.
4
Click  Add.
5
In the table, enter the following settings:
Parameter name
Parameter value list
Parameter unit
pw (Power)
range(300,200,2000)
W
6
From the Run continuation for list, choose No parameter.
7
From the Reuse solution from previous step list, choose Yes.
8
In the Study toolbar, click  Compute.
In the following create plots for the gas temperature, H mole fraction, electron temperature, ion densities, neutral densities, and relative ionization rates.
Results
Gas Temperature and H Mole Fraction
1
In the Results toolbar, click  1D Plot Group.
2
In the Settings window for 1D Plot Group, type Gas Temperature and H Mole Fraction in the Label text field.
Global 1
1
Right-click Gas Temperature and H Mole Fraction and choose Global.
2
In the Settings window for Global, locate the y-Axis Data section.
3
In the table, enter the following settings:
Expression
Unit
Description
plas.T
K
Temperature
4
Locate the x-Axis Data section. From the Parameter list, choose Expression.
5
In the Expression text field, type pw/plas.Vol_gm.
6
In the Unit field, type W/cm^3.
Global 2
1
Right-click Global 1 and choose Duplicate.
2
In the Settings window for Global, locate the y-Axis Data section.
3
In the table, enter the following settings:
Expression
Unit
Description
plas.x_wH
1
Mole fraction
Gas Temperature and H Mole Fraction
1
In the Model Builder window, click Gas Temperature and H Mole Fraction.
2
In the Settings window for 1D Plot Group, click to expand the Title section.
3
From the Title type list, choose Label.
4
Locate the Plot Settings section.
5
Select the x-axis label checkbox. In the associated text field, type Power density (W/cm<sup>3</sup>).
6
Select the Two y-axes checkbox.
7
In the table, select the Plot on secondary y-axis checkbox for Global 2.
8
Locate the Legend section. From the Position list, choose Upper left.
Ions Density
1
In the Results toolbar, click  1D Plot Group.
2
In the Settings window for 1D Plot Group, type Ions Density in the Label text field.
Global 1
1
Right-click Ions Density and choose Global.
2
In the Settings window for Global, locate the y-Axis Data section.
3
In the table, enter the following settings:
Expression
Unit
Description
plas.n_wH_1p
1/cm^3
Number density
plas.n_wH2_1p
1/cm^3
Number density
plas.n_wH3_1p
1/cm^3
Number density
4
Locate the x-Axis Data section. From the Parameter list, choose Expression.
5
In the Expression text field, type pw/plas.Vol_gm.
6
In the Unit field, type W/cm^3.
7
Click to expand the Legends section. Find the Include subsection. Clear the Description checkbox.
8
Select the Expression checkbox.
9
In the Ions Density toolbar, click  Plot.
Ions Density
1
In the Model Builder window, click Ions Density.
2
In the Settings window for 1D Plot Group, locate the Title section.
3
From the Title type list, choose Label.
4
Locate the Plot Settings section.
5
Select the x-axis label checkbox. In the associated text field, type Power density (W/cm<sup>3</sup>).
6
Locate the Axis section. Select the y-axis log scale checkbox.
7
Select the Manual axis limits checkbox.
8
In the y minimum text field, type 1e4.
9
In the y maximum text field, type 2e12.
10
Locate the Legend section. From the Position list, choose Middle right.
Electron Temperature
1
In the Results toolbar, click  1D Plot Group.
2
In the Settings window for 1D Plot Group, type Electron Temperature in the Label text field.
Global 1
1
Right-click Electron Temperature and choose Global.
2
In the Settings window for Global, locate the y-Axis Data section.
3
In the table, enter the following settings:
Expression
Unit
Description
plas.Te
V
Maxwellian
4
Locate the x-Axis Data section. From the Parameter list, choose Expression.
5
In the Expression text field, type pw/plas.Vol_gm.
6
In the Unit field, type W/cm^3.
7
In the Electron Temperature toolbar, click  Plot.
Electron Temperature
1
In the Model Builder window, click Electron Temperature.
2
In the Settings window for 1D Plot Group, locate the Title section.
3
From the Title type list, choose Label.
4
Locate the Plot Settings section.
5
Select the x-axis label checkbox. In the associated text field, type Power density (W/cm<sup>3</sup>).
6
Select the y-axis label checkbox. In the associated text field, type Electron temperature (eV).
7
Locate the Axis section. Select the Manual axis limits checkbox.
8
In the y minimum text field, type 0.
9
In the y maximum text field, type 2.
10
In the Electron Temperature toolbar, click  Plot.
Neutrals Density
1
In the Results toolbar, click  1D Plot Group.
2
In the Settings window for 1D Plot Group, type Neutrals Density in the Label text field.
Global 1
1
Right-click Neutrals Density and choose Global.
2
In the Settings window for Global, locate the y-Axis Data section.
3
In the table, enter the following settings:
Expression
Unit
Description
plas.n_wH
1/m^3
Number density
plas.n_wH2
1/m^3
Number density
plas.n_wHn2
1/m^3
Number density
plas.n_wHn3
1/m^3
Number density
4
Locate the x-Axis Data section. From the Parameter list, choose Expression.
5
In the Expression text field, type pw/plas.Vol_gm.
6
In the Unit field, type W/cm^3.
7
Locate the Legends section. Find the Include subsection. Clear the Description checkbox.
8
Select the Expression checkbox.
9
In the Neutrals Density toolbar, click  Plot.
Neutrals Density
1
In the Model Builder window, click Neutrals Density.
2
In the Settings window for 1D Plot Group, locate the Title section.
3
From the Title type list, choose Label.
4
Locate the Plot Settings section.
5
Select the x-axis label checkbox. In the associated text field, type Power density (W/cm<sup>3</sup>).
6
Locate the Axis section. Select the y-axis log scale checkbox.
7
Locate the Legend section. From the Position list, choose Middle right.
Relative Ionization Rates
1
In the Results toolbar, click  1D Plot Group.
2
In the Settings window for 1D Plot Group, type Relative Ionization Rates in the Label text field.
Global 1
1
Right-click Relative Ionization Rates and choose Global.
2
In the Settings window for Global, locate the y-Axis Data section.
3
In the table, enter the following settings:
Expression
Unit
Description
(plas.r_rg3_1+plas.r_rg3_2)/(plas.r_rg3_1+plas.r_rg3_2+plas.r_rg3_3+plas.r_rg2_21+plas.r_rg1_7)
1
Hn2,3+H2=>H3++e
(plas.r_rg3_3+plas.r_rg2_21)/(plas.r_rg3_1+plas.r_rg3_2+plas.r_rg3_3+plas.r_rg2_21+plas.r_rg1_7)
1
e+H2=>2e+H2+ & H2+H2+=>H3++H
plas.r_rg1_7/(plas.r_rg3_1+plas.r_rg3_2+plas.r_rg3_3+plas.r_rg2_21+plas.r_rg1_7)
1
e+H=>H++2e
4
Locate the x-Axis Data section. From the Parameter list, choose Expression.
5
In the Expression text field, type pw/plas.Vol_gm.
6
In the Unit field, type W/cm^3.
7
Locate the Legends section. Find the Include subsection. Clear the Solution checkbox.
8
In the Relative Ionization Rates toolbar, click  Plot.
Relative Ionization Rates
1
In the Model Builder window, click Relative Ionization Rates.
2
In the Settings window for 1D Plot Group, locate the Title section.
3
From the Title type list, choose Label.
4
Locate the Plot Settings section.
5
Select the x-axis label checkbox. In the associated text field, type Power density (W/cm<sup>3</sup>).
6
Select the y-axis label checkbox. In the associated text field, type Relative ionization rate.
7
Locate the Axis section. Select the y-axis log scale checkbox.
8
Select the Manual axis limits checkbox.
9
In the y minimum text field, type 1e-5.
10
In the y maximum text field, type 10.
11
Locate the Legend section. From the Position list, choose Lower left.
In the next part of this example the global model is solved self-consistently with the Boltzmann equation in the two-term approximation.
Choose to solve the Boltzmann equation in the two-term approximation in the HF limit and set the reduced angular frequency. Reduce the maximum energy of the EEDF domain (this value is found from trial-and-error and verifying that the high energy region of the EEDF is weekly populated for values larger than 50 eV).
Before solving the global model and the Boltzmann equation it is necessary to obtain initial conditions for the EEDF. This is done with the EEDF Initialization study that only solves for the EEDF. This solution is used as initial condition to a subsequent study.
Plasma (plas)
1
In the Model Builder window, collapse the Component 1 (comp1) > Plasma (plas) node.
2
In the Model Builder window, click Plasma (plas).
3
In the Settings window for Plasma, locate the Electron Energy Distribution Function Settings section.
4
From the Electron energy distribution function list, choose Boltzmann equation, two-term approximation (linear).
5
Select the Oscillating field checkbox.
6
In the ω/N text field, type w0/plas.Nn.
7
In the εmax text field, type 50[V].
Add Study
1
In the Home toolbar, click  Add Study to open the Add Study window.
2
Go to the Add Study window.
3
Find the Studies subsection. In the Select Study tree, select Preset Studies for Selected Physics Interfaces > EEDF Initialization.
4
Click the Add Study button in the window toolbar.
5
In the Select Study tree, select General Studies > Stationary.
6
Click the Add Study button in the window toolbar.
7
In the Home toolbar, click  Add Study to close the Add Study window.
Study 2
In the Study toolbar, click  Compute.
Results
EEDF Initialization
In the Settings window for 1D Plot Group, type EEDF Initialization in the Label text field.
The next study solves the global and the Boltzmann equation in the two-term approximation fully coupled. The power is parameterized from 300 to 2000 W. The EEDF previously computed is used as initial conditions.
Study 3
Step 1: Stationary
1
In the Model Builder window, under Study 3 click Step 1: Stationary.
2
In the Settings window for Stationary, click to expand the Values of Dependent Variables section.
3
Find the Initial values of variables solved for subsection. From the Settings list, choose User controlled.
4
From the Method list, choose Solution.
5
From the Study list, choose Study 2, EEDF Initialization.
6
Locate the Study Extensions section. Select the Auxiliary sweep checkbox.
7
Click  Add.
8
In the table, enter the following settings:
Parameter name
Parameter value list
Parameter unit
pw (Power)
range(300,200,2000)
W
9
From the Run continuation for list, choose No parameter.
10
From the Reuse solution from previous step list, choose Yes.
11
In the Model Builder window, click Study 3.
12
In the Settings window for Study, locate the Study Settings section.
13
Clear the Generate default plots checkbox.
14
In the Study toolbar, click  Compute.
15
In the Label text field, type Boltzmann EEDF.
16
In the Model Builder window, collapse the Boltzmann EEDF node.
Initialization EEDF
1
In the Model Builder window, click Study 2.
2
In the Settings window for Study, type Initialization EEDF in the Label text field.
Maxwellian EEDF
1
In the Model Builder window, click Study 1.
2
In the Settings window for Study, type Maxwellian EEDF in the Label text field.
3
In the Model Builder window, collapse the Maxwellian EEDF node.
Initialization EEDF
In the Model Builder window, collapse the Initialization EEDF node.
In the same plots prepared before add the results when solving for the EEDF.
Results
Global 3
1
In the Model Builder window, under Results > Gas Temperature and H Mole Fraction right-click Global 1 and choose Duplicate.
2
In the Settings window for Global, locate the Data section.
3
From the Dataset list, choose Boltzmann EEDF/Solution 3 (sol3).
4
Click to expand the Coloring and Style section. Find the Line style subsection. From the Line list, choose Dashed.
5
From the Color list, choose Cycle (reset).
6
Locate the Legends section. Clear the Show legends checkbox.
Global 4
1
In the Model Builder window, under Results > Gas Temperature and H Mole Fraction right-click Global 2 and choose Duplicate.
2
In the Settings window for Global, locate the Data section.
3
From the Dataset list, choose Boltzmann EEDF/Solution 3 (sol3).
4
Locate the Coloring and Style section. Find the Line style subsection. From the Line list, choose Dashed.
5
Locate the Legends section. Clear the Show legends checkbox.
Global 2
1
In the Model Builder window, under Results > Ions Density right-click Global 1 and choose Duplicate.
2
In the Settings window for Global, locate the Data section.
3
From the Dataset list, choose Boltzmann EEDF/Solution 3 (sol3).
4
Locate the Coloring and Style section. Find the Line style subsection. From the Line list, choose Dashed.
5
From the Color list, choose Cycle (reset).
6
Locate the Legends section. Clear the Show legends checkbox.
7
In the Ions Density toolbar, click  Plot.
Global 2
1
In the Model Builder window, under Results > Electron Temperature right-click Global 1 and choose Duplicate.
2
In the Settings window for Global, locate the Data section.
3
From the Dataset list, choose Boltzmann EEDF/Solution 3 (sol3).
4
Locate the y-Axis Data section. In the table, enter the following settings:
Expression
Unit
Description
plas.Te
V
Boltzmann
5
In the Electron Temperature toolbar, click  Plot.
Global 2
1
In the Model Builder window, under Results > Neutrals Density right-click Global 1 and choose Duplicate.
2
In the Settings window for Global, locate the Data section.
3
From the Dataset list, choose Boltzmann EEDF/Solution 3 (sol3).
4
Locate the Coloring and Style section. Find the Line style subsection. From the Line list, choose Dashed.
5
From the Color list, choose Cycle (reset).
6
Locate the Legends section. Clear the Show legends checkbox.
Global 2
1
In the Model Builder window, under Results > Relative Ionization Rates right-click Global 1 and choose Duplicate.
2
In the Settings window for Global, locate the Data section.
3
From the Dataset list, choose Boltzmann EEDF/Solution 3 (sol3).
4
Locate the Coloring and Style section. Find the Line style subsection. From the Line list, choose Dashed.
5
From the Color list, choose Cycle (reset).
6
Locate the Legends section. Clear the Show legends checkbox.
Prepare a plot to show the computed EEDF for the different applied powers.
Boltzmann EEDF/Solution 3 (5) (sol3)
1
In the Model Builder window, expand the Results > Datasets node.
2
Right-click Results > Datasets > Boltzmann EEDF/Solution 3 (sol3) and choose Duplicate.
3
In the Settings window for Solution, locate the Solution section.
4
From the Component list, choose Extra Dimension from Plasma (plas_eedf_xdim).
EEDF Boltzmann
1
In the Results toolbar, click  1D Plot Group.
2
In the Settings window for 1D Plot Group, locate the Data section.
3
From the Dataset list, choose Boltzmann EEDF/Solution 3 (5) (sol3).
4
In the Label text field, type EEDF Boltzmann.
Line Graph 1
1
Right-click EEDF Boltzmann and choose Line Graph.
2
In the Settings window for Line Graph, locate the Selection section.
3
From the Selection list, choose All domains.
4
Locate the y-Axis Data section. In the Expression text field, type plas.fcap.
5
Locate the x-Axis Data section. From the Parameter list, choose Expression.
6
In the Expression text field, type plas.xeedf.
7
Click to expand the Legends section. Select the Show legends checkbox.
8
From the Legends list, choose Evaluated.
9
In the Legend text field, type eval(plas.Pgm/plas.Vol_gm, W/cm^3) W/cm<sup>3</sup>.
10
In the EEDF Boltzmann toolbar, click  Plot.
Line Graph 2
1
Right-click Line Graph 1 and choose Duplicate.
2
In the Settings window for Line Graph, locate the Data section.
3
From the Dataset list, choose Boltzmann EEDF/Solution 3 (5) (sol3).
4
From the Parameter selection (pw) list, choose Last.
5
Locate the y-Axis Data section. In the Expression text field, type plas.fmax.
6
Click to expand the Coloring and Style section. From the Color list, choose Black.
7
Locate the Legends section. From the Legends list, choose Manual.
8
In the table, enter the following settings:
Legends
Maxwellian
EEDF Boltzmann
1
In the Model Builder window, click EEDF Boltzmann.
2
In the Settings window for 1D Plot Group, locate the Title section.
3
From the Title type list, choose Label.
4
Locate the Plot Settings section.
5
Select the x-axis label checkbox. In the associated text field, type Energy (eV).
6
Select the y-axis label checkbox. In the associated text field, type EEDF (eV <sup>-3/2</sup>).
7
Locate the Axis section. Select the y-axis log scale checkbox.
8
Select the Manual axis limits checkbox.
9
In the x minimum text field, type 0.
10
In the x maximum text field, type 25.
11
In the y minimum text field, type 1e-9.
12
In the y maximum text field, type 2.
13
In the EEDF Boltzmann toolbar, click  Plot.