Global Model Coupled with the Two-Term Boltzmann Equation

The electron energy distribution function (EEDF) plays an important role in the overall behavior of discharges. In this work, the formation period of an argon plasma with special attention for the EEDF is studied. The plasma is created within a 4 cm gap by a DC source voltage of 1 kV in series with a 100 kΩ resistance at 100 mTorr (compare with the Micro-cathode sustained discharge in Ar example from Ref. 1). A global model in the local field approximation is used to describe the temporal evolution of the plasma species. The rate coefficients for electron impact reactions and electron mobility are obtained from suitable integration of cross sections over the EEDF, and the EEDF is computed from the steady state Boltzmann equation in the two-term approximation (B2T) Ref. 2. The computed EEDF is influenced by how electrons gain energy from the electric field and lose it afterward in collisions with the background gas.

Coupling the B2T with a space dependent model is very computational 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.

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 or can be described using an analytic model. 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. In the following, we make use of the fast computational time to investigate the plasma evolution during the formation period through to steady state.

When using a plasma global 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 total mass or pressure into account, the mass-continuity equation can also be solved

The electron number density is obtained from electroneutrality

and if 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 a EEDF. The EEDF can be either analytic or can be 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 reduce electric field must be given as input to the Boltzmann solver and the electron mean energy comes from averaging over the computed EEDF.

In this work the LFA is used and the B2T is solved. The reduced electric field given as input to the B2T comes from the circuit equation

where Vp is the plasma potential, VDC is the applied voltage, and R is the circuit series resistance. The plasma current Ip is computed from

where e is the electron charge, A is the plasma cross section area, μN is the reduced electron mobility, E/N is the reduced electric field, and N is the gas density. Solving for E/N one obtains

where d is the gap distance. One can anticipate that an increase in the plasma density will reduce the electric field in the discharge gap.

plasma chemistry

The plasma chemistry suggested in Ref. 1 is used and presented in Table 1. The chemical mechanism consists of 5 species and 11 reactions.

Table 1: table of collisions and reactions modeled.
reaction	formula	type
1	e+Ar=>e+Ar	Momentum	0
2	e+Ar=>e+Ars	Excitation	11.5
3	e+Ars=>e+Ar	Excitation	-11.5
4	e+Ar=>2e+Ar+	Ionization	15.8
5	e+Ars=>2e+Ar+	Ionization	4.427
6	2e+Ar+=>Ar+e	Excitation	-15.8
7	e+Ar2+=>Ars+e	Excitation	-3
8	2Ar+Ar+=>Ar+Ar2	3-body Ar2+ creation
9	Ars+2Ar=>3Ar	Quenching
10	Ars+Ars=>e+Ar2+	Penning ionization
11	Ar+Ar2+=>2Ar+Ar+	Charge transfer

The model also includes the surface reactions presented in Table 2.

Table 2: table of surface reactions.
reaction	formula	Sticking Coefficient
1	Ar+=>Ar	1
2	Ar2+=>2Ar	1
3	Ars=>Ar	1

The neutral excited species revert to ground state with sticking coefficient equal to one.

Results and Discussion

Figure 1 shows computed and analytic EEDFs for a mean electron energy of 5 eV. These results were obtained with the EEDF Initialization study that only solves for the Boltzmann equation in the two-term approximation without solving for the degrees of freedom from the global model. This approach allows to have a first insight of the EEDF and observe if an analytic EEDF is accurate enough for the modeling expectations. From Figure 1 it is possible to see that above 20 eV the computed EEDF drops much faster than the Maxwellian EEDF. As a consequence the Maxwellian EEDF overpredicts electron impact processes with high energy thresholds like ionization from the ground state. The Druyvesteyn EEDF is in much better agreement with the computed EEDF and it might be good enough for some models.

From Figure 2 to Figure 5, simulation results are presented for the global model coupled with the B2T. Figure 2 shows the temporal evolution of the plasma species and the reduced field in the plasma. In the first instants there is no plasma in the gap and the electric field maintains a constant value until the plasma creation begins. With the plasma creation the current in the gap (which is only plasma conduction current in this model) increases and the voltage decreases as presented in Figure 3. At steady state the plasma is sustained with only 30 V in the gap that corresponds to a field of 3.5 Td.

From Figure 4 it is possible to observe that the temporal evolution of the electron mean energy (obtained by averaging the computed EEDF) follows the same trend as the electric field. The electron mean energy decreases with the electric field and undershoots before reaching the steady state. The same behavior can be inferred from the temporal evolution of the EEDFs presented in Figure 5. Further analyses of the EEDF allows to gain a deeper understating of the plasma physics. In the beginning the electrons have a large population above 15 eV as it is necessary to facilitate the plasma breakdown. After the plasma formation, and due to the decrease of the electric field, the electron population cools down and the EEDF develops a tail with a steeper slope. As time progresses, and with the increase of the argon excited state density, the influence of the superelastic reactions in the EEDF becomes noticeable with the appearance of a bump at the high energy end.

Figure 1: Computed and analytic EEDFs for a mean electron energy of 5 eV.

Figure 2: Temporal evolution of the plasma species and the reduced electric field.

Figure 3: Temporal evolution of the plasma voltage and current.

Figure 4: Temporal evolution of the mean electron energy.

Figure 5: Computed EEDF at several instants of the simulation.

References

1. S. Pancheshnyi, B. Eismann, G.J.M. Hagelaar, and L.C. Pitchford, Computer code ZDPlasKin, http://www.zdplaskin.laplace.univ-tlse.fr (University of Toulouse, LAPLACE, CNRS-UPS-INP, Toulouse, France, 2008).

2. 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.

3. Phelps database, www.lxcat.net, retrieved 2017.

Plasma_Module/Global_Modeling/boltzmann_global_model_argon

Modeling Instructions

From the menu, choose .

New

In the window, click

Model Wizard

In the window, Select the interface and the study.

click

In the tree, select .

Click .

Click

In the tree, select .

Click

Geometry 1

Create a geometry to define the plasma volume and contact surfaces.

Rectangle 1 (r1)

In the toolbar, click

In the window for , locate the section.

In the text field, type rad.

In the text field, type gap.

Add some parameters to define the geometry, background gas temperature, and circuit components.

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
rad	0.4[cm]	0.004 m
gap	0.4[cm]	0.004 m
Res	1e5[ohm]	1E5 Ω
Tg	300[K]	300 K
P0	100[torr]	13332 Pa
Vdc	1000[V]	1000 V

Add some variables to define the diffusion length, the reduced electric field in the plasma, and the plasma current and voltage.

Definitions

Variables 1

In the window, under right-click and choose .

In the window for , locate the section.

In the table, enter the following settings:


Name	Expression	Unit	Description
Ldiff	((pi/gap)^2+(2.405/rad)^2)^-0.5	m
Ip	e_constAreaplas.neplas.ENplas.muN
Vp	Vdc-Res*Ip
EN	Vdc/(gap+Rese_constAreaplas.nenojac(plas.muN)/plas.Nn)/plas.Nn
Area	pi*rad^2	m²