Update of LoKI-B simulation tool with electron density growth by electron-impact ionizations Duarte Nuno Barreto Gonçalves Thesis to obtain the Master of Science Degree in Engineering Physics Supervisor: Prof. Luís Paulo da Mota Capitão Lemos Alves Examination Committee Chairperson: Prof. João Pedro Saraiva Bizarro Supervisor: Prof. Luís Paulo da Mota Capitão Lemos Alves Members of the Committee: Prof. Vasco António Dinis Leitão Guerra Dr. Antonio Tejero Del Caz September 2017
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Update of LoKI-B simulation tool with electron densitygrowth by electron-impact ionizations
Duarte Nuno Barreto Gonçalves
Thesis to obtain the Master of Science Degree in
Engineering Physics
Supervisor: Prof. Luís Paulo da Mota Capitão Lemos Alves
Examination Committee
Chairperson: Prof. João Pedro Saraiva BizarroSupervisor: Prof. Luís Paulo da Mota Capitão Lemos AlvesMembers of the Committee: Prof. Vasco António Dinis Leitão Guerra
Dr. Antonio Tejero Del Caz
September 2017
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Acknowledgements
I would like to acknowledge the guidance of my supervisor Prof. Luís Lemos Alves. The opportunity to
be part of the project KIT-PLASMEBA and liberty to explore the subject further, resulted in a gratifying
and interesting work.
I thank the team of this project, for even the brief discussions helped me understand and consolidate
the study. I genuinely thank Antonio Tejero that was always available to discuss and explain matters
whenever I asked.
A special thanks to my family and friends for their support through life.
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Este trabalho foi financiado pela Fundação para a Ciência e
Tecnologia, através do Projecto PTDC/FIS-PLA/1243/2014
(KIT-PLASMEBA) e pelas bolsas BL136/2016_IST-ID e
BL150/2017_IST-ID.
This work has been supported by the portuguese Fundação para a
Ciência e Tecnologia, under Project PTDC/FIS-PLA/1243/2014
(KIT-PLASMEBA) and grants BL136/2016_IST-ID and
BL150/2017_IST-ID.
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Resumo
Os plasmas de baixa temperatura têm vindo a ser cada vez mais usados em aplicações industriais,
sendo que nos últimos anos houve um desenvolvimento de aplicações ambientais e biológicas. Ex-
iste portanto uma necessidade tecnológica para melhorar a previsibilidade do comportamento destes
plasmas. Neste contexto, o projecto KIT-PLASMEBA pretende desenvolver novas ferramentas, uma
das quais um programa computacional LisbOn KInetics (LoKI), que contém um modelo de resolução
numérica da equação de Boltzmann para eletrões (LoKI-B).
O objectivo deste trabalho é introduzir um novo tratamento das ionizações por impacto eletrónico,
contribuindo para o desenvolvimento do LoKI-B. Para este fim, criou-se uma nova rotina de ionização,
onde foram incluídos dois modelos de crescimento de densidade eletrónica, bem como um operador de
ionização que usa uma secção eficaz diferencial de ionização. De forma a integrar completamente esta
rotina no código LoKI-B, foi realizado um acoplamento com a rotina de colisões eletrão-eletrão.
As previsões do primeiro coeficiente de ionização de Townsend melhoraram significativamente para
Árgon, sendo que para Azoto molecular as previsões do LoKI estão agora dentro das incertezas ex-
perimentais. Foram efectuadas verificações com outro código de resolução numérica da equação de
Boltzmann para eletrões, onde se comprovou a viabilidade do trabalho efectuado. Uma análise dos
vários operadores colisionais de ionização, permitiu descrever os mecanismos pelos quais a ionização
por impacto electrónico influencia a função de distribuição dos eletrões.
Palavras chave: plasmas de baixa temperatura, ionização por impacto electrónico,LoKI-B, modelização cinética
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Abstract
Low-temperature plasmas have been extensively used in industrial applications, with developments in
environmental and biological applications being made in recent years. There is then a technological
need to improve the predictability on the behaviour of these plasmas. To this end, the project KIT-
PLASMEBA aims at the development of new tools, one of them being the kinetic code LisbOn KInetics
(LoKI), which contains an electron Boltzmann equation solver (LoKI-B).
The goal of this work is to introduce a new description of electron-impact ionizations, supporting the
development of LoKI-B. To this end, a new ionization routine was created in which two electron density
growth models were included, as well as a non-conservative ionization collisional operator that uses a
differential ionization cross section. In order to seamlessly integrate this routine with LoKI-B, a coupling
with the electron-electron collisions routine was made.
Predictions for the first Townsend ionization coefficient improved significantly for the case of Argon,
and into experimental data uncertainty in the case of molecular Nitrogen. Comparisons against another
electron Boltzmann equation solver, verified the accuracy of the present work. An analysis of the various
ionization collisional operators allowed a description of the various mechanism for which electron-impact
ionization influences the electron distribution function.
Here the first term account for the scattered electrons, that enter the distribution function with energy
between u and u + du, produced by a primary electron with energy u + VI ; the second term accounts
for the primary electrons that leave the distribution function due to ionizing collisions; and the third term
accounts for secondary electrons that enter the distribution function with zero energy. Note that 2.18
corresponds to the non-conservative form of 2.16.
These two predefined energy sharing modes will be used mostly for benchmark purposes.2The value 4 may lead to some confusion since an ionizing collision produces two electrons, not four. This value comes from
a change of variable. Assume that we have an electron with energy ε, belonging to the electron distribution. The number of new
electrons Ne produced by this primary electron with energy between ε and ε+ dε, after an ionizing collision is given by
BNe(ε) dε = 2 εσI (ε) f(ε) dε,
in which B is a variable that ensures the correct units for the expression. To write this result as function of the product electrons’
Changing to a kinetic-energy description on equation 2.21 and using the relations in 2.9 we have
∂F (~r , ~v , t)
∂t= N
neN v
me
4πe
[√u
1
ne
∂ne∂t
f (u) +√u
1
ne
∂ne∂t
f10 (u)P1 +
√u
(1
ne
∂ne∂t
+ j ω
)f1
1 (u)P1 (u) ej w t]
= γ
u CI√2eume
f (u) + uCI√
2eume
f10 (u)P1 + u
CI√2eume
+ jωR√
2eume
f11 (u)P1 e
j w t
,in which CI is the ionization rate coefficient.
Spatial dependent term
Assuming a density variation along the Z direction, the gradient term is
∇~r ·[~v F (~r,~v, t)
]≈ ~v · ∇~r [ne(~r, t)F (~v, t)] = ~v · ~ez
∂ne(~r, t)
∂zF (~v, t) = v
∂ne(~r, t)
∂zF (~v, t) cos θ.
22
Using the two term approximation,
∇~r ·[~v F (~r,~v, t)
]≈v ∂ne(~r, t)
∂zF (~v, t) cos θ
=v∂ne(~r, t)
∂z
[F 0(v) +
(F 1
0 (v) + F 11 (v)ej ω t
)cos θ
]cos θ
=v∂ne(~r, t)
∂z
[F 0(v) cos θ +
(F 1
0 (v) + F 11 (v)ej ω t
)cos2 θ
]and by writting cos θ and cos2 θ in terms of the Legendre polynomials
=v∂ne(~r, t)
∂z
[F 0(v)P1 +
(F 1
0 (v) + F 11 (v)ej ω t
)(2P2 + P0
3
)]≈v ∂ne(~r, t)
∂z
(F 1
0 (v) +F 1
1 (v)ej ω t
3P0 + F 0(v)P1
)≈v ∂ne(~r, t)
∂z
(F 1
0 (v)
3+ F 0(v) cos θ
)
If we assume an electron density exponential spatial growth (using 2.19), the gradient term becomes
∇~r ·[~v F (~r,~v, t)
]= v αne(~r, t)
(F 1
0 (v)
3+ F 0(v) cos θ
),
and by changing to a kinetic-energy description and using the relations in 2.9 we have
∇~r ·[~v F (~r,~v, t)
]= γ
[αR u
(f1
0 (u)
3+ f(u) cos θ
)].
Electric field term
Assuming that the applied electric field is along the zz
~E = −E~ez,
and knowing that
(∂
∂vz
)vx,vy
= cosθ∂
∂v+sin2θ
v
∂
∂cosθ,
then the electric field term for a general component of order l of the distribution function expansion is
−e ~E(t)
me· ∇~v F = ne
∑l=0
−e ~E(t)
me· ∇~v(F lPl) = ne
∑l=0
eE(t)
me
∂F l
∂vPl cosθ + ne
∑l=0
eE(t)
me
F l
v
∂Pl∂cosθ
sin2θ,
and using relations 2.4 and 2.5
−e ~E(t)
me· ∇~v F = ne
eE(t)
me
[∑l=0
∂F l
∂v
(l + 1)Pl+1 + l Pl−1
2l + 1+∑l=0
F l
v
l(l + 1)
2 l + 1(Pl−1 − Pl+1)
]=
= neeE(t)
me
[∑l=−1
(∂F l+1
∂v
l + 1
2 l + 3+F l+1
v
(l + 1)(l + 2)
2 l + 3
)Pl +
∑l=1
(∂F l−1
∂v
l
2 l − 1− F l−1
v
l(l − 1)
2l − 1
)Pl
]
For the two-term expansion 2.2 we have
23
−e ~E(t)
me· ∇~v F = ne
1
3 v2
∂
∂v
(eE(t)
meF 1(v, t) v2
)P0 + ne
∂
∂v
(eE(t)
meF 0(v)
)P1. (2.22)
Changing to a kinetic-energy description on equation 2.22, and using relations 2.7 for the Fourier
expansion, and 2.9, we obtain
∇~v ·
(−e ~E(t)
meF (~r,~v, t)
)
=γ
∂
∂u
[u3ER(t)f1(u, t)
]+ u
∂
∂u[ER(t) f(u)] cos θ
=γ
∂
∂u
[u3
(E0Rf
10 (u) + E1R
f11 (u)
2
)]+ u
∂
∂u
[E0R f(u) + E1R f(u) ejωt
]cos θ
+ (2.23)
+γ∂
∂u
[u3
(
:0E1Rf
10 (u)ejωt +
:0E0Rf
11 (u)ejωt
)]=γ
∂
∂u
[u3
(E0Rf
10 (u) + E1R
f11 (u)
2
)]+ u
∂
∂u
[E0R f(u) + E1R f(u) ejωt
]cos θ
.
The last two terms of 2.23 are zero since with HF electric field the stationary anisotropic part (f10 ) is
zero, and with DC electric field the time dependent anisotropic part (f11 ) is zero.
Elastic collision term
The elastic collisional operator may be written as [29]
I0 =me
M
1
v2
∂
∂v
[v3 νc
(ne(~r, t)F
0(v) +kB Tgme v
∂ne(~r, t)F0(v)
∂v
)],
in which νc = Nσc(v)v, with σc = 2π∫ π
0(1−cosχ)σ(v, χ) sinχdχ is the momentum transfer cross-section,
σ the elastic collision cross-section, and χ the angle between the initial and final electron trajectories.
We can see that the second term ensures that the electron distribution function goes to equilibrium with
the gas molecules at temperature Tg, when the elastic collisional operator dominates. This form was
first deduced by Chapmann and Cowling [30].
Changing to a kinetic-energy description on equation 2.24 and using the relations in 2.9 we have
I0 = γ∂
∂u
[2me
Mu2 σc(u)
(f(u) +
kB Tge
∂f(u)
∂u
)]. (2.24)
Inelastic/superelastic collision term
The inelastic/superelastic conservative collisions between electrons and neutrals can lead to the excitation/de-
excitation of electronic, vibrational and rotational states of the gas. The operator follows the same struc-
ture for all types of collisions. The subscripts i and j denominate the electron energy levels of the
molecule, being j the most energetic one. The inelastic/superelastic term can be written as [26],
24
J0 = ne∑i,j
v + vij
vνij (v + vij)F
0(v + vij)− νij(v)F 0(v)
+v − vij
vνji (v − vij)F 0(v − vij)− νji(v)F 0(v)
(2.25)
We define ni as the density of the particles in state i, the corresponding relative population being
δi = ni/N , and u ± uij = 12me
2 (v + vij)2 as the electron energy plus or minus the energy interval
between levels i and j.
Changing to a kinetic-energy description on equation 2.25 and using the relations in 2.9 we have
J0
γ=∑i,j
δi(u+ uij
)σij(u+ uij) f(u+ uij)− δiu σij(u) f(u)
+
+∑i,j
δj(u− uij
)σji(u− uij) f(u− uij)− δju σji(u) f(u)
. (2.26)
It is now possible to use Klein-Rosseland relation
gju σji(u) = gi(u+ uij
)σij(u+ uij),
in which gi and gj are the statistical weights of energy levels i and j respectively, to re-write equation
2.26 as
J0
γ=∑i,j
δi(u+ uij
)σij(u+ uij) f(u+ uij)− δiu σij(u) f(u)
+
+∑i,j
gigjδju σij(u) f(u− uij)−
gigjδj(u+ uij
)σij(u+ uij) f(u)
.
Rotational collision term - continuous approximation
Rotational collisions have energy thresholds much smaller than the vibrational and electronic ones.
Hence the numerical code must use an energy step smaller than the lowest threshold, which is compu-
tationally demanding. In the case of N2, O2 and H2, it is possible to use a continuous approximation for
the rotational collisional operator, with a "Chapman-Cowling" term [31],
J0rot
γ=
∂
∂u
[4Bσ0u
(f(u) +
kBTge
∂f(u)
∂u
)].
Anisotropic collisional operator
We will assume an effective collision frequency that accounts for elastic, inelastic and super-elastic
processes, to write the anisotropic collisional operator as,
I1 + J1 = −νeff ne(~r, t)F 1,
or in a kinetic-energy description
I1 + J1 = −γ u σeff(u) f1(u).
25
2.4.2 Isotropic and anisotropic components of the electron of Boltzmann equa-
tion
Using the normalization condition 2.2.1, the isotropic component of the EBE can be obtain by integrating
the equation multiplied by the first Legendre polynomial P0 using the corresponding orthogonality relation
2.3. The anisotropic component can be obtained by integrating the EBE multiplied by P1. Since we will
only use a DC or a HF electric field, we will also separate the stationary/non-stationary anisotropic parts
in each case. The different terms can be grouped as,
Zero order terms
γ4πuCI√
2eume
f(u)+γ4π
(αR u
f10 (u)
3
)+ γ4π
∂
∂u
[u
3
(E0Rf
10 (u) + E1R
f11 (u)
2
)]=
=γ4π∂
∂u
[2me
Mu2 σc(u)
(f(u) +
kB Tge
∂f(u)
∂u
)]+ 4π
[J0(u) + JI(u) + J0
rot(u)]
(2.27)
First order stationary terms
γ4π
3u
CI√2eume
f10 (u) + γ
4π
3αR u f(u) + γ
4π
3u∂
∂u(E0Rf(u)) = −γ 4π
3uσeff (u)f1
0 (u) (2.28)
First order non-stationary terms
γ4π
3u
CI√2eume
+ jωR√
2eume
f11 (u) + γ
4π
3u∂
∂u(E1Rf(u)) = −γ 4π
3uσeff (u)f1
1 (u) (2.29)
The full isotropic Boltzmann equation
We are interested mostly on the isotropic part of the electron distribution function, termed EEDF. We can
get the equation for the EEDF by rewriting equations 2.28 on f10 and 2.29 on f1
1 (u) in order to f(u), and
substituting them in equation 2.27.
First we re-write the anisotropic, Fourier order 1, equation of f11 (u). Using σIeff (u) = σeff (u) + CI√
2eume
we have,
f11 (u)
σIeff (u) + jωR√
2eume
= −∂ (E1Rf(u))
∂u
⇔ f11 (u) = −
σIeff (u)− jωR/√
2eume
σIeff (u)2
+ ω2R/
2eume
∂ (E1Rf(u))
∂u
since we need only the real part
f11 (u) = −
σIeff (u)
σIeff (u)2
+ ω2R/
2eume
E1R∂f(u)
∂u. (2.30)
26
Secondly lets re-write the anisotropic, Fourier order 0, equation of f10 (u),
f10 (u)σIeff (u) = −αRf(u)− E0R
∂f(u)
∂u
⇔ f10 (u) = − αR
σIeff (u)f(u)− E0R
σIeff (u)
∂f(u)
∂u. (2.31)
We now use these results in equation 2.27 to obtain the equation for the EEDF,
−u CI√2eume
f(u) +u3
α2R
σeff (u)f(u) +
u3
αRE0R
σeff (u)
∂f(u)
∂u+
+∂
∂u
[u
3
E0R
σIeff (u)
(αRf(u) + E0R
∂f(u)
∂u
)]+∂
∂u
(u
3E2
1R
σIeff (u)/2
σIeff (u)2
+ ω2R/
2eume
∂f(u)
∂u
)+
+∂
∂u
[2me
Mu2 σc(u)
(f(u) +
kB Tge
∂f(u)
∂u
)]+J0(u)
γ+JI(u)
γ+J0rot(u)
γ= 0 (2.32)
Usually, LoKI will be working with either the exponential temporal growth or the exponential spatial
growth of the electron density, and also with either a DC or a HF electric field. So four equations can
be obtained from 2.32. However, a spatial electron density growth is not realistic in a HF field since the
mean electron drift velocity is zero, meaning that an exponential spatial electron density profile, created
by electron-impact ionizations, is not plausible. 3
2.4.3 Temporal growth with DC electric field
−u CI√2eume
f(u) +∂
∂u
(u
3
E20R
σIeff (u)
∂f(u)
∂u
)+∂
∂u
[2me
Mu2 σc(u)
(f(u) +
kB Tge
∂f(u)
∂u
)]+
+J0(u)
γ+JI(u)
γ+J0rot(u)
γ= 0
2.4.4 Temporal growth with HF electric field
−u CI√2eume
f(u) +∂
∂u
(u
3
E21R
2
σIeff (u)
σIeff (u)2
+ ω2R/
2eume
∂f(u)
∂u
)+∂
∂u
[2me
Mu2 σc(u)
(f(u) +
kB Tge
∂f(u)
∂u
)]+
+J0(u)
γ+JI(u)
γ+J0rot(u)
γ= 0
2.4.5 Spatial growth with DC electric field
αR3
[u
σeff
(αRf(u)+E0R
∂f(u)
∂u
)+∂
∂u
(uE0R
σefff(u)
)]+∂
∂u
(u
3
E20R
σeff (u)
∂f(u)
∂u
)+
+∂
∂u
[2me
Mu2 σc(u)
(f(u) +
kB Tge
∂f(u)
∂u
)]+J0(u)
γ+JI(u)
γ+J0rot(u)
γ= 0
3 It is important to note that in LoKI, the electric field value is the measured one, not the field’s amplitude, this corresponds to
the effective RMS electric field Eeff1R = E1R√
2. However, during this formulation, the value kept is the field’s amplitude.
27
2.5 Conservation and transport equations
2.5.1 Drift-diffusion equation
The electron drift-diffusion equation is a particle transport equation that describes the various phenom-
ena responsible for electron motion. In gas discharges, usually this equation has a drift term, due to the
effect of the electric field, and a diffusion term, due to the effect of an electron density gradient,
~Γ = −∇r(Dne)− µ0ne ~E,
with the electron flux Γ is defined as,
~Γ = ne~vdrift. (2.33)
With the two-term Boltzmann equation, the electron flux can be calculated from the anisotropic part of
the distribution function
~Γ = ne~vdrift =
∫ ∞0
ne~v~F 1(v) cos θd~v
= ne
∫ ∞0
∫ π
0
∫ 2π
0
v3F 1(v) cos2 θ sin θdθdφdv ~ez
=4πne
3
∫ ∞0
v3F 1(v)dv ~ez
=ne3
√2e
me
∫ ∞0
uf1(u)du ~ez.
Since the electron distribution function was expanded in Fourier series, the electron flux will also be
expanded,
~Γ = ~Γ0 + ~Γ1 ej ω t.
Temporal or spatial growth of the electron density with DC electric field
Since we are in the DC case, we will use f10 (u). Although we will use either the spatial growth, or
temporal growth of the electron density, we will calculate the electron flux for both models simultaneously.
Using equation 2.31, with αR = 1neN
∂ne
∂z ,
ne3
√2e
me
∫ ∞0
uf10 (u)du = −ne
3
√2e
me
∫ ∞0
u
(1
neN
∂ne∂z
1
σIeff (u)f(u) +
E0R
σIeff (u)
∂f(u)
∂u
)du
= − 1
3N
√2e
me
∫ ∞0
u1
σIeff (u)f(u)du
∂ne∂z− ne
1
3N
√2e
me
∫ ∞0
u1
σIeff (u)
∂f(u)
∂uduE0
= −D∇rne + neµ0E0
Thus we have obtained the drift-diffusion equation,
Γ0 = −D∇rne + neµ0E, (2.34)
28
with the following expressions for the electron diffusion coefficient and mobility
D =1
3N
√2e
me
∫ ∞0
u1
σIeff (u)f(u)du (2.35)
µ0 = − 1
3N
√2e
me
∫ ∞0
u1
σIeff (u)
∂f(u)
∂udu. (2.36)
In the case of the exponential spatial growth, the diffusion and mobility coefficients have the same
form, with the quantity σIeff defined as,
σIeff (u) = σeff (u).
In the case of the exponential temporal growth, the drift-diffusion equation is simply
Γ0 = neµ0E,
since there is no electron density gradient. The effect of electron ionization collisions is then included in
the quantity σIeff defined as,
σIeff (u) = σeff (u) + CI
√me
2eu. 4
Temporal growth of the electron density with HF electric field
In the HF case the electric field oscillates with a high frequency ω. This means that the drift velocity will
now have an oscillatory term. In the HF case there is no DC field and as a result the DC component of
the electron drift velocity is zero,
Γ0 = 0.
The time-dependent term for electron flux is,
Γ1 =ne3
√2e
me
∫ ∞0
uf11 (u)du.
Using equation 2.30,
ne3
√2e
me
∫ ∞0
uf11 (u) = −ne
3
√2e
me
∫ ∞0
uσIeff (u)− jωR
√me
2eu
σIeff (u)2
+ ω2R/
2eume
E1R∂f(u)
∂u
= −ne1
3N
√2e
me
∫ ∞0
uσIeff (u)− jωR
√me
2eu
σIeff (u)2
+ ω2R/
2eume
∂f(u)
∂uE1
= neµ1E1,
and the drift-diffusion equation is
Γ1 = neµ1E1,
4A usual doubt arises from the fact that σeff already accounts for the ionization cross section and it seems that there is no need
to add the CI term. However, the CI coefficient comes from the electron density growth term, ∂ne∂t
and not from the collisional
operator.
29
with the HF mobility given by
µ1 = µr + j µi,
µr = − 1
3N
√2e
me
∫ ∞0
uσIeff (u)
σIeff (u)2
+ ω2R/
2eume
∂f(u)
∂u
µi =1
3N
√2e
me
∫ ∞0
uωR√
me
2eu
σIeff (u)2
+ ω2R/
2eume
∂f(u)
∂u.
Again, the effect of the temporal growth due to ionization collisions appears on the σIeff quantity.
2.5.2 Particle balance equation
In this subsection the calculation of particle balance will be made by calculating the zeroth velocity
moment of the EBE.
Looking back to equation 2.32 we will integrate it, term by term, in reading order.
Time variation term
−∫ ∞
0
uCI
√me
2 e uf(u)du = −CI
√me
2 e
∫ ∞0
f(u)√udu = −CI
√me
2 e,
in which we have used the normalization condition 2.2.1.
Space variation term ∫ ∞0
αR3
[u
σeff
(αRf(u) + E0R
∂f(u)
∂u
)]du
The first part of this integral gives,∫ ∞0
αR3
u
σeffαRf(u)du =
αRne
1
3N
∫ ∞0
u
σeff
∂ne∂z
f(u)du =
=αRne
√me
2 eD∇rne,
on which we have used 2.35. The second part of the integral gives,∫ ∞0
αR3
u
σeffE0R
∂f(u)
∂udu = αR
1
3N
∫ ∞0
u
σeff
∂f(u)
∂uduE0 =
= −αRne
√me
2 eneµ0E0,
on which we have used 2.36. So the integration of the spatial variation terms is
αRne
√me
2 eD∇rne −
αRne
√me
2 eneµ0E0 =
√me
2 e
αRne
(−nevdrift) = −αRvdrift√me
2 e.
Electric field term
.
Electron density exponential temporal growth with DC field∫ ∞0
∂
∂u
(u
3
E20R
σIeff (u)
∂f(u)
∂u
)du =
[u
3
E20R
σIeff (u)
∂f(u)
∂u
]∞0
= 0
30
Electron density exponential spatial growth with DC field
∫ ∞0
∂
∂u
[uE0R
3σeff
(αRf(u) + E0R
∂f(u)
∂u
)]du =
[uE0R
3σeff
(αRf(u) + E0R
∂f(u)
∂u
)]∞0
= 0
Electron density exponential temporal growth with HF field∫ ∞0
∂
∂u
(u
3E2R
σIeff (u)/2
σIeff (u)2
+ ω2R/
2eume
∂f(u)
∂u
)du =
[u
3E2R
σIeff (u)/2
σIeff (u)2
+ ω2R/
2eume
∂f(u)
∂u
]∞0
= 0
Elastic collision term
∫ ∞0
∂
∂u
[2me
Mu2 σc(u)
(f(u) +
kB Tge
∂f(u)
∂u
)]du =
[2me
Mu2 σc(u)
(f(u) +
kB Tge
∂f(u)
∂u
)]∞0
= 0
Inelastic/superelastic collision term
Using the operator before applying the Klein-Rosseland relation, that is, with explicit superelastic cross
sections,
∫ ∞0
J0
γdu =
∫ ∞0
∑i,j
1st term︷ ︸︸ ︷
δi(u+ uij
)σij(u+ uij) f(u+ uij)−
2ndterm︷ ︸︸ ︷δiu σij(u) f(u)
du+
+
∫ ∞0
∑i,j
3rdterm︷ ︸︸ ︷
δj(u− uij
)σji(u− uij) f(u− uij)−
4thterm︷ ︸︸ ︷δju σji(u) f(u)
du .
First term ∫ ∞0
δi(u+ uij
)σij(u+ uij) f(u+ uij) du, ε = u+ uij and dε = du
=
∫ ∞uij
δiε σij(ε) f(ε) dε =
∫ ∞0
δiε σij(ε) f(ε) dε−∫ uij
0
δiε*0
σij(ε) f(ε) dε︸ ︷︷ ︸σij(u)=0, if u<uij
=
=δi
√me
2 e
〈νij(u)〉N
= δi
√me
2 eCij
Second term
−∫ ∞
0
δiu σij(u) f(u)du = −δi√me
2 e
〈νij(u)〉N
= −δi√me
2 eCij
Third term∫ ∞0
δj(u− uij
)σji(u− uii) f(u− uij) du, ε = u− uij and dε = du
=
∫ ∞−uij
δjε σji(ε) f(ε) dε =
∫ ∞0
δjε σji(ε) f(ε) dε+
∫ 0
−uij
δjε*0
σji(ε) f(ε) dε︸ ︷︷ ︸σij(u)=0, if u<uij
=
=δj
√me
2 e
〈νji(u)〉N
= δj
√me
2 eCji
31
Fourth term
−∫ ∞
0
δju σji(u) f(u)du = −δj√me
2 e
〈νji(u)〉N
= −δj√me
2 eCji
In the end we have, ∫ ∞0
J0
γ(u)du = 0
Ionization collision term∫ ∞0
JIγdu =
∫ ∞0
∫ ∞2u+VI
εqisec(ε, u)f(ε)dε+
∫ 2u+VI
u+VI
εqisca(ε, u)f(ε)dε− uσI(u)f(u)
du
First term∫ ∞0
∫ ∞2u+VI
εqisec(ε, u)f(ε)dεdu =
∫ ∞0
∫ (ε−VI)/2
0
εqisec(ε, u)f(ε)dudε =
∫ ∞0
εσI(ε)f(ε)dε =
√me
2 e
〈νI〉N
,
in which we have used relation 2.11. Second term∫ ∞0
∫ 2u+vI
u+VI
εqisca(ε, u)f(ε)dεdu =
∫ ∞0
∫ ε−VI
(ε−VI)/2
εqisca(ε, u)f(ε)dudε, u′ = ε− VI − u
=
∫ ∞0
∫ 0
(ε−VI)/2
εqisca(ε, ε− VI − u′)f(ε)(−du′)dε =
∫ ∞0
∫ (ε−VI)/2
0
εqisec(ε, u′)f(ε)du′dε =
=
∫ ∞0
εσI(ε)f(ε)dε =
√me
2 e
〈νI〉N
,
in which we have used relation 2.11. Third term∫ ∞0
−uσI(u)f(u)du = −√me
2e
〈νI〉N
.
The sum of all terms is then, ∫ ∞0
JI(u)
γdu =
√me
2e
〈νI〉N
Particle balance equation
The particle balance equation for an exponential temporal growth of the electron density is
CI + 0 =〈νI〉N
⇔ ∂ne∂t
+∂Γ
∂z= S,
in which it is possible to see that since the time variation of the electron density is equal to the ionization
rate (source term).
The particle balance equation for an exponential spatial growth of the electron density is
0 + αRvdrift =〈νI〉N
⇔ ∂ne∂t
+∂Γ
∂z= S.
Here the spatial variation of the electron flux is equal to the ionization source term. The number of
particles is also conserved, since the reduced Townsend ionization coefficient is defined as
α =〈νI〉vdrift
.
32
2.5.3 Energy balance equation
In this subsection the calculation of the energy balance equation will be made by calculating the second
velocity moment of the EBE, i.e. by multiplying the EBE by u and integrating in all energies.
This calculation is helpful in understanding the different sources of energy sharing. In LoKI, the
power balance equation is used to check the quality of the numerical calculations.
The following will be obtained in terms of the energy density and the energy flux,
nE = ne
∫ ∞0
uf(u)√udu = ne 〈u〉
ΓE =
√2 e
me
ne3
∫ ∞0
u2f10 (u)du,
writing the energy flux in terms of its diffusion and mobility components
ΓE = −DE∇rnE + nEµEE0 (2.37)
DE =
√2 e
me
1
3N
1
〈u〉
∫ ∞0
u2
σefff(u)du
µE = −√
2 e
me
1
3N
1
〈u〉
∫ ∞0
u2
σeff
∂
∂uf(u)du
This representation of the energy flux is somewhat unusual. Some authors prefer this formulation
due to its consistency with the two-term EBE and the fact that 2.37 uses transport parameters that can
be calculated from integration over the EEDF [16].
Time variation term
ne
∫ ∞0
u−CI√
2eume
f(u)u du = −ne CI√me
2 e
∫ ∞0
u f(u)√udu = −CI
√me
2 enE
Space variation term
ne
∫ ∞0
αR3
[u2
σeff
(αRf(u) + E0R
∂f(u)
∂u
)]du =
=αR3N
(∫ ∞0
u2
σefff(u)du
)∂ne∂z
+αR3N
ne
(∫ ∞0
u2
σeff
∂f(u)
∂udu
)E0
=αR
√me
2 e(DE∇rne 〈u〉 − ne 〈u〉µEE0)
=αR
√me
2 e(DE∇rnE − nEµEE0)
=αR
√me
2 e(−ΓE)
33
Electric field term
Electron density exponential temporal growth with DC field
ne
∫ ∞0
u∂
∂u
(u
3
E20R
σIeff (u)
∂f(u)
∂u
)du = − ne
3N
∫ ∞0
u
σIeff
∂f(u)
∂uduE0E0R
= +
√me
2 eneµ0E0E0R
=
√me
2 e(Γ)E0R
=
√me
2 ene vdriftE0R
Electron density exponential temporal growth with HF field,
ne
∫ ∞0
u∂
∂u
(u
3E2
1R
σIeff (u)/2
σIeff (u)2
+ ω2R/
2eume
∂f(u)
∂u
)du =− ne
3N
∫ ∞0
uσIeff (u)/2
σIeff (u)2
+ ω2R/
2eume
∂f(u)
∂uduE1
E1R
2
=
√me
2 eneµrE1
E1R
2
=
√me
2 e< (Γ1)
E1R
2
=
√me
2 ene<(vdrift)
E1R
2.
Electron density exponential spatial growth with DC field
ne
∫ ∞0
u∂
∂u
[uE0R
3σeff
(αRf(u) + E0R
∂f(u)
∂u
)]du =
=−1
3N
[∫ ∞0
u
σefff(u)du
∂ne∂z
+ ne
∫ ∞0
u
σeff
∂f(u)
∂uduE0
]E0R
=−√me
2 e(D∇rne − neµ0E0)E0R
=
√me
2 enevdriftE0R
In all of these cases, the power density (in eV cm−3s−1), gained from the electric field, can be written,
apart from a constant, as
PE = ~J · ~E = −~Γ · ~E = ΓE.
which corresponds to a Joule heating term.
Elastic collision term
ne
∫ ∞0
u∂
∂u
[2me
Mu2σc(u)
(f(u) +
kBTge
∂f(u)
∂u
)]du = −ne
2me
M
∫ ∞0
u2σc(u)
(f(u) +
kBTge
∂f(u)
∂u
)du
A suggestion to have a more intuitive insight of this power is to assume a Maxwellian distribution at
the electron temperature Te. Then the Chapmann and Cowling term is of the order −Tg/Te, and the
power lost in elastic collisions can be written approximately as [29],
−ne2me
M
√me
2 e〈u〉Cc
(1− Tg
Te
),
34
with Cc the rate coefficient of elastic collisions. As expected, the power lost to elastic collisions is zero if
Te = Tg.
Inelastic/superelastic collision term
The inelastic/superelastic collision term is composed by 4 terms and the the power is written for each
term separately.
First term∫ ∞0
δiu(u+ uij
)σij(u+ uij)f(u+ uij)du = , using ε = u+ uij
= δi
∫ ∞uij
(ε− uij
)εσij(ε)f(ε) dε = δi
∫ ∞0
(ε− uij
)εσij(ε)f(ε) dε− δi
∫ uij
0
(ε− uij
)ε
*0σij(ε)f(ε) dε︸ ︷︷ ︸
σij(u)=0, if u<uij
=
= δi
∫ ∞0
ε2σij(ε)f(u) dε− δi∫ ∞
0
uijεσij(ε)f(ε) dε =
= δi
√me
2 e
∫ ∞0
εCij(ε)f(ε)√ε dε− δi
√me
2 euij
∫ ∞0
Cij(ε)f(ε)√ε dε =
= δi
√me
2 e
[〈εCij(ε)〉 − uijCij
].
Second term
−∫ ∞
0
δiu2σij(u)f(u)du = −δi
√me
2 e〈uCij(u)〉 .
Analogous calculations could be made for the third and fourth term. In the end, summing all four
terms we have,
ne
∫ ∞0
uJ0(u)
γdu = ne
√me
2 e
∑ij
uij ( δjCji − δiCij )
This can be seen as the power gained and the power lost due to superelastic and inelastic collisions,
respectively.
Ionization collision term
ne
∫ ∞0
JI(u)
γdu = ne
∫ ∞0
u
∫ ∞2u+VI
εqisec(ε, u)f(ε)dε+
∫ 2u+VI
u+VI
εqisca(ε, u)f(ε)dε− uσI(u)f(u)
du
Starting with the second term,∫ ∞0
∫ 2u+vI
u+VI
uεqisca(ε, u)f(ε)dεdu =
∫ ∞0
∫ ε−VI
(ε−VI)/2
u ε qisca(ε, u)f(ε)dudε, u′ = ε− VI − u∫ ∞0
∫ 0
(ε−VI)/2
(ε− VI − u′)εqisca(ε, ε− VI − u′)f(ε)(−du′)dε =
∫ ∞0
∫ (ε−VI)/2
0
(ε− VI − u′)εqisec(ε, u′)f(ε)du′dε =
=
∫ ∞0
∫ ∞2u′+VI
(ε− VI − u′)εqisec(ε, u′)f(ε)dεdu′.
35
Now summing the first two terms we have,∫ ∞0
u
∫ ∞2u+VI
εqisec(ε, u)f(ε)dεdu+
∫ ∞0
u
∫ 2u+VI
u+VI
εqisca(ε, u)f(ε)dεdu =
=
∫ ∞0
u
∫ ∞2u+VI
εqisec(ε, u)f(ε)dεdu+
∫ ∞0
∫ ∞2u′+VI
(ε− VI − u′)εqisec(ε, u′)f(ε)dεdu′ =
=
∫ ∞0
∫ ∞2u′+VI
(ε− VI)εqisec(ε, u′)f(ε)dεdu′ =
∫ ∞0
∫ (ε−VI)/2
0
(ε− VI)εqisec(ε, u′)f(ε)du′dε
=
∫ ∞0
(ε− VI) ε σI(ε)f(ε)dε,
in which we have used relation 2.11. Finally, adding the third term,
ne
∫ ∞0
uJI(u)
γdu = ne
∫ ∞0
(ε− VI) ε σI(ε)f(ε)dε− ne∫ ∞
0
u2σI(u)f(u)du =
= −neVI∫ ∞
0
ε σI(ε)f(ε)dε
= −ne VI√me
2 eCI
Energy balance equation
For the exponential temporal growth model with DC field, the energy balance equation is,
CInE + 0− ne vdriftE0R = −ne2me
M
∫ ∞0
u2σc(u)
(f(u) +
kBTge
∂f(u)
∂u
)du+
+ne∑ij
uij (njCji − niCij )− ne VICI ,
which is similar to an energy transport equation
∂nE∂t
+∂ΓE∂z︸ ︷︷ ︸
here is zero
−ΓE0 = −Pel + Psup − Pinel − PI
with Γ the electron particle flux, ΓE the electron energy flux, Pel the power lost in elastic collisions, Psup
and Pinel the power gained and lost in superelastic/inelastic collisions, respectively, and PI the power
lost in electron-impact ionizations.
The energy balance equation for the case of the exponential temporal growth model with a HF
field is very similar, except that here we take into account the real part of the electron drift velocity
<(vdrift) = <(>
0v0 + v1e
jωt
)= v1 cos(ωt).
For the exponential spatial growth model the energy balance equation is,
0 + αRΓE − ne vdriftE0R = −ne2me
M
∫ ∞0
u2σc(u)
(f(u) +
kBTge
∂f(u)
∂u
)du+
+ne∑ij
uij (njCji − niCij )− ne VICI ,
which is similar to an energy transport equation,
∂nE∂t︸ ︷︷ ︸
here is zero
+∂ΓE∂z− ΓE0 = −Pel + Psup − Pinel − PI
36
Chapter 3
Computational Approach
3.1 Solving the Boltzmann equation
In LoKI the purpose is to solve equation 2.32, in order to obtain the EEDF, and then to calculate the
transport parameters and the rate coefficients needed on the Chemistry module.
Equation 2.32 can be written, detailing its functional dependence on the kinetic energy u, through f
and its derivatives
B
(u, f(u),
df(u)
du,d2f(u)
du2
)= 0.
In low-temperature plasmas, the probability of having electrons with energy much higher than the mean
electron energy is low. In consequence, the value of the EEDF at these energies will also be very low,
and we can calculate the electron coefficients using EEDF values between a certain range of electron
energies, with negligible errors.
Within this range it is then possible to define a discrete 1D energy-grid, with say N small energy
intervals, where the discretized equations is converted in a set of N equations
B1
(f1,
df1
du,d2f1
du2
)= 0
B2
(f2,
df2
du,d2f2
du2
)= 0
(...)
Bk
(fk,
dfkdu
,d2fkdu2
)= 0
(...)
BN−1
(fN−1,
dfN−1
du,d2fN−1
du2
)= 0
BN
(fN ,
dfNdu
,d2fNdu2
)= 0.
However there are still 3N unknown variables (f , dfdu and d2fdu2 ) on these N equations. In order to have
a solvable system, it is necessary to linearise the EEDF derivatives, for example by approximating them
37
with finite differences. This will lead to a system of equations Bk equations that are a function of the
EEDF at the different sites of the grid
Bk [(...), fk−2, fk−1, fk, fk+1, fk=2, (...)] = 0.
The result is a homogeneous system of N equations with N unknown variables, for which only the
solution that follows the normalization condition, equation 2.8, is the physical one. If the Bk terms are
linear in f , then the full system to be numerically solved is,
∑m
Bk,mfm = 0
∑k
fk√uk = 1.
The matrix B will be called EBE matrix throughout this work. Solving this system is the discrete
equivalent of solving equation 2.32.
If all EBE terms are linear on the EEDF, it is possible to add the normalization condition to a B matrix
line, rendering the system well defined and solvable. In LoKI, the normalization condition is added to the
first line of the matrix, and the system to solve becomes
B1,1 +√u1 B1,2 +
√u2 B1,3 +
√u3 · · · B1,N +
√uN
B2,1 B2,2 B2,3 · · · B2,N
B3,1 B3,2 B3,3 · · · B2,N
......
.... . .
...
BN,1 BN,2 BN,2 · · · BN,N
·
f1
f2
f3
...
fN
=
1
0
0
...
0
.
However, if secondary electrons are included, someBkm terms are non-linear on the EEDF. This non-
linear system needs to be linearised and then iterated until convergence on the non-linear coefficients
and the EEDF.
3.2 Discretization of the electron Boltzmann equation
In order to numerically solve the EBE a discretization of its different terms needs to be made.
These terms can be classified as linear terms on the EEDF, terms with integrals on the EEDF, and
terms that involve EEDF derivatives. Linear terms can be discretized with a conversion rule that makes
the connection between an energy on the continuous plane and on the discrete grid. Terms that involve
integrals can be discretized with a quadrature rule, here the simple mid-point (or rectangle) rule will
be used. Finally, terms with EEDF derivatives will be discretized using a finite difference method, here
centred differences will be used.
38
This discretization will be made using an energy grid that ranges from zero to a maximum energy
umax, divided in N points. The energy step is defined as ∆u = umax/N , with the energy at each point k
defined as u+k = k∆u. Quantities that are a function of the electron energy are defined at the interval’s
limit, for example, the cross-section σ+effK
= σeff (u+k ). The EEDF will be defined at the middle of the
interval
fk = f(uk)
in which uk =(k − 1
2
)∆u. Thus, the quadrature rule and finite difference method are applied with the
goal of arriving at expressions with the EEDF defined at the middle of the energy interval.
The details of the calculations performed in this subsection are given on the Annex A.
3.2.1 Linear terms on the electron energy distribution function
Linear terms are discretized directly by writing them at the middle of the energy interval. These terms are
the electron-density-growth terms, both spatial and temporal, and the conservative inelastic operators.
Time variation term
The electron density exponential temporal growth term is, for an energy uk = (k − 1/2)∆u,u CI√2 e ume
f(u)
u=uk
≡√m
2eCI fk
√uk.
Space variation term
One of the three terms that constitute the electron density exponential spatial growth term is linear on
the EEDF. It can be discretized for an energy uk,[α2R
3
u
σeff (u)f(u)
]u=uk
≡ α2R
3
ukσeffK
fk
Conservative inelastic/superelastic collisions term
For the conservative inelastic collisional operator, the discretization is[J0(u)
γ
]u=uk
≡∑i,j
[δi uk+mij σ
ijk+mij fk+mij − δi uk σijk fk +
gigjδj uk σ
ijk fk−mij −
gigjδj uk+mij σ
ijk+mij fk
]
in which mij defined as mij = floor(uij
∆u
)the lowest closer number of energy intervals that cor-
respond to the energy level difference (uij) of the excitation/de-excitation of the inelastic/superelastic
collisions.
39
3.2.2 Terms with integrals on the electron energy distribution function
Using the mid-point quadrature rule the discrete ionization operator for a specific interval of energy
u = uk,[JI(u)
γ
]u=uk
≡n∑
j=2k+MI+1
ujqisec(uj , uk)fj∆u+
2k+MI∑j=k+MI+1
ujqisec(uj , uj−k−MI
)fj∆u−ukfk(k−MI)/2∑
j=1
qisec(uk, uj)∆u.
in which MI is defined as MI = round(VI
∆u
), the closest number of energy intervals corresponding to
the ionization energy.
3.2.3 Terms with derivatives on the electron energy distribution function
For these terms a centred finite differences method will be used. There are many variations of this
method, our set of rules being described in Annex A.
Electron density exponential spatial growth terms
There are two of the three electron density exponential spatial growth terms that include EEDF’s deriva-
tive. The discretization for an energy interval u = uk is,[αR3
u
σeffE0R
∂f(u)
∂u
]u=uk
=αR3
ukσeffK ∆u
E0R
(fk+1 − fk−1
2
),
[αR3
∂
∂u
(u
σeffE0R f(u)
)]u=uk
=αRE0R
6
[fk+1
k
σ+effK
+ fk
(k
σ+effK
− k − 1
σ+effK−1
)− fk−1
k − 1
σ+effK−1
].
Rotational collisions term - continuous approximation
We will treat separately the two terms that compose the rotational continuous approximation collision
operator, ∂
∂u[4Bσ0uf(u)]
u=uk
≡ 2Bσ0 (fk+1k + fk − fk−1(k − 1)) ,
∂
∂u
[4Bσ0u
kBTge
∂f(u)
∂u
]u=uk
≡ 4Bσ0kBTge∆u
[fk+1k − fk(2k − 1) + fk−1(k − 1)] .
Elastic collisions term
The elastic collision operator can also be divided in two terms, with first order and second-order deriva-
tives, respectively∂
∂u
[2me
Mu2σc(u)f(u)
]u=uk
≡ me
M
fk+1k
2σ+c k∆u+ fk
[k2σ+
c k − (k − 1)2σ+c k−1
]∆u− fk−1(k − 1)2σ+
c k−1∆u,
∂
∂u
[2me
Mu2σc(u)
kBTge
∂f(u)
∂u
]u=uk
≡ 2me
M
kBTge
[fk+1k
2σ+ck − fk
(k2σ+
ck + (k − 1)2σ+c k−1
)+ fk−1(k − 1)2σ+
c k−1
].
40
Electric field term
The electric field term can be discretized for an energy u = uk,
∂
∂u
((E
N
)2u
3g(u)
∂f
∂u
)=
(E
N
)21
3∆u
[fk+1 k g
+k − fk(k g+
k + (k − 1) g+k−1) + fk−1 (k − 1) g+
k−1
],
with g+k being
g+k =
1CI√me√
2ek∆u+ σ+
eff k
× 1
1 +ω2
Rme
(CI√me+
√2ek∆u σ+
effk)2
.
Matrix form for the terms with EEDF derivatives
We can rearrange the terms with EEDF derivatives, with the exception of the here presented first term
of the spatial variation, to get a set of expressions on fk, fk−1 and fk+1,
Ak−1 fk−1 − (Ak +Bk) fk +Bk+1 fk+1,
with
Ak =
(E
N
)2k
3∆ug+k +
me
Mk2σ+
c k
[2kBTge−∆u
]+ 2Bσ0
[2kBTge
k
∆u− k]
+αRE0R
6
k
σeff k
Bk =
(E
N
)2k − 1
3∆ug+k−1 +
me
M(k − 1)2σ+
c k−1
[2kBTge
+ ∆u
]+ 2Bσ0
[2kBTge
k − 1
∆u+ (k − 1)
]+αRE0R
6
k − 1
σeff (k−1)
3.3 Solving the non-linear electron Boltzmann equation
On the previous subsection we presented the discretization of the EBE, with some terms involving CI
and α. These terms are calculated using the EEDF, hence they are non-linear. This means that they
have to be explicitly calculated, to then be used when writing the EBE matrix. This needs to be done
iteratively until CI and α convergence is achieved. Each of the electron-density growth mechanisms is
linearized in a different way, although the iterative algorithms are similar.
3.3.1 Convergence over the ionization rate or first Townsend coefficients
Using the exponential temporal growth model as example, the non-linear EBE to be solved is the dis-
cretized version of,
−u CI√2eume
f(u) +∂
∂u
(u
3E2
1R
σIeff (u)/2
σIeff (u)2
+ ω2R/
2eume
∂f(u)
∂u
)+I0(u)
γ+J0(u)
γ+JI(u)
γ+J0rot(u)
γ= 0.
In this equation, the ionization rate coefficient appears both on the time derivative and on the electric
field term through σIeff . By knowing CI we can solve this equation as a linear system and obtain the
EEDF. However the ionization rate coefficient is calculated as,
CI =
√2 e
me
∫ ∞0
uσI(u)f(u)du,
41
meaning that it is necessary to know the EEDF to calculate this coefficient.
The system is then non-linear. In order to solve it, an iterative method is necessary, with CI playing
the role of convergence parameter, being calculated in successive iterations until convergence.
3.3.2 Iterative algorithm
Instead of using a numerical method for non-linear algebraic systems, a simple convergence cycle was
used. Initially the EEDF is calculated assuming no electron-density growth model. With this first estimate
of the EEDF, the first Townsend coefficient, or the ionization rate coefficient are calculated. A new EBE
matrix is then constructed with terms that depend on these coefficients. Solving the resulting equations
system as a linear system, a new estimate of the EEDF is obtained.
The cycle can be described by:
1. An EEDF is calculated without including secondary electrons;
2. Using the previous EEDF as an initial guess, the convergence parameter is calculated. The conver-
gence parameters is the ionization rate coefficient (temporal growth model) or the first Townsend
ionization coefficient (spatial growth model);
3. Using the previous value of the convergence parameters, the EBE matrix is constructed now in-
cluding the production of secondary electrons due to ionization events;
4. The ionization routine cycle commences with a new calculation of the EEDF;
5. The new EEDF is used to update the convergence parameters;
6. The convergence criteria are checked. If convergence is not achieved the cycle continues back to
step 3.
Once the convergence criteria are met, usually corresponding to relative differences smaller than
10−10 for the convergence parameters and the EEDF, the cycle is terminated. A flow chart of this
iterative algorithm can bee seen in Figure 3.1. This simple method is stable for most working conditions.
3.3.3 Coupling with electron-electron collisions
The electron density growth due to electron-impact ionizations is not the only non-linear mechanism
included on the EBE. In particular, LoKI-B has also a routine for electron-electron collisions, in which
some of the coefficients on the electron-electron collisional operator (Jee) contain integrals over the
EEDF. When the different non-linear mechanism are activated simultaneously, we need to find an ade-
quate numerical algorithm that ensures convergence.
In order to couple two non-linear routines it is important to take into account some aspects. First it is
important not to try to converge too many parameters at the same time, since their values might start tp
oscillate between iterations. Second, there must be some communication between the routines so that
they both converge to the same solution.
The coupling was made as follows:
42
1. The EEDF is calculated without accounting for secondary electrons;
2. The ionization routine is performed without accounting for electron-electron collisions;
3. After the calculation of the ionization convergence parameters, the electron-electron collisions
routine is performed;
• Inside this routine, the terms due to electron-impact ionization are included but they are not
updated between iterations;
• The ionization convergence parameter is calculated at the end of each cycle;
4. After convergence on the electron-electron collisions routine, the convergence criteria are applied
to the ionization convergence parameters;
5. If the previous criteria are not satisfied, the ionization routine is performed again now including, but
not updating, electron-electron collisions terms;
6. The global cycle is repeated continuing from step 3.
A graphical description of this iterative algorithm can be seen in Figure 3.2.
The e-e collisions routine converges if the ratio of electric field’s power to electron-electron collision’s
power is bigger than 107. This global convergence can be hard to achieve with:
• Very-high frequency values:
• Large numbers of energy grid-points (∼5000 more);
• Very high ionization degree;
• Spatial electron density growth model.
It is sometimes hard to predict if a certain set of conditions will allow the convergence.
Electron density temporal growth model
In the case of the electron density temporal growth model, the EBE to be solved is the discretized version
of
−u CI√2eume
f(u)+∂
∂u
(u
3
E21R
2
σIeff (u)
σIeff (u)2
+ ω2R/
2eume
∂f(u)
∂u
)+I0(u)
γ+Jrot(u)
γ+J0(u)
γ+JI(u)
γ+Jee(u)
γ= 0,
for the HF case, and an equivalent form for the DC case by setting ω = 0 and E1R/2 → E0R. In
solving this equation, the rate coefficients CI will be used as convergence parameter. The ionization
rate coefficient appears on the term due to the time dependence, and on the electric field term since
σIeff (u) = σeff (u) +CI√
2eume
,
and both terms need to be updated at each iteration, following changes in CI .
43
After the calculation of the initial EEDF estimate, in which secondary electrons are not accounted
for, the program proceeds to calculate the ionization rate coefficient. By definition the ionization rate
coefficient is,
CI =
√2 e
me
∫ ∞0
uσI(u)f(u)du.
Initially the calculation of the ionization rate coefficient was done with this expression. However, there
were some particle balance errors and significant power balance errors when a final solution was ob-
tained. At the time the power balance errors seemed to be proportional to the energy step, which
indicated that it may be due to some discretization error. Indeed an error introduced by the discretization
was identified on the ionization collisional operator when doing analytical verifications of the discretized
EBE. In order to obtain a better correspondence between CI and the ionization operator, the calculation
of CI was made by integrating the corresponding operator over the energy grid,
CI =
√2 e
me
n∑k=1
JI
∣∣∣k∆u. (3.1)
This procedure ensures a perfect verification of the numerical particle balance equation, and minimizes
the relative error (keeping it bellow 10−12) in the calculation of the numerical energy balance equations.
Since the initial EEDF estimate is done without accounting for secondary electrons, there can be
large differences between the initial and the final values of CI . A large difference between consecutive
coefficients, may cause the EEDF or CI to overshoot (or oscillate), in a way that convergence is difficult
or impossible to obtain. In order to ensure that the convergence parameter converges slowly, a mixing
of solutions was used. Let C0I be the value used on the EBE matrix on the previous iteration, and CxI its
current value calculated with equation 3.1. The CI value to be used, when writing the next EBE matrix,
is
CI = x CxI + (1− x) C0I ,
with x ∈ [0; 1] being the mixing parameter. Usually, we have adopted x = 0.7 as the mixing parameter
for the temporal electron density growth.
Having a new CI estimate, the temporal growth term and the electric field term can be updated. A
new EEDF is calculated, a new value is assigned to C0I , a new calculation of CxI is performed, and the
next CI is calculated with mixing of solutions. The process goes on until convergence.
Convergence tests are performed after the calculation of the new CI value and before updating the
EBE matrix. Since CI works as a convergence parameter, the first test is on the relative difference
between the previous and the current ionization rate coefficient,
||CI − C0I ||
C0I
< 10−10.
If this criterion is met, a second test is performed upon the EEDF, calculating the maximum relative
difference between corresponding values of the consecutive EEDFs,
max
(||fnew − fold||
fold
)< 10−10.
44
Electron density spatial growth model
With exponential spatial growth model for the electron density, the EBE to be solved is the discretized
form of
αR3
[u
σeff
(αRf(u) + E0R
∂f(u)
∂u
)+
∂
∂u
(uE0R
σefff(u)
)]+
∂
∂u
(u
3
E20R
σeff (u)
∂f(u)
∂u
)+
+I0(u)
γ+Jrot(u)
γ+J0(u)
γ+JI(u)
γ+Jee(u)
γ= 0.
Here, the convergence parameter is the reduced first Townsend ionization coefficient, of which calcula-
tion is not as direct as for the ionization rate coefficient.
The first Townsend coefficient α was defined in this work, see sec 2.3.4, as
α =∇rnene
· ~ez.
corresponding to a spatial growth frequency. The reduced coefficient relates to the net electron produc-
tion as [16]
αR =CIvdrift
, (3.2)
in which the ionization rate coefficient is calculated with the same expression 3.1 as in the temporal
growth model case, and the electron drift velocity is calculated using equations 2.34 and 2.33. Writing
this drift-diffusion equation in respect to the drift velocity we have,
vdrift = −N DαR + µ0E0,
which leads to a quadratic equation on αR,
α2RND − αRµ0E0 + CI = 0,
with solution
αR =µ0E0 −
√(µ0E0)
2 − 4NDCI
2ND, (3.3)
in which µ0 and D are calculated using
D =
√2 e
me
1
3N
n∑k=1
ukσeff k
fk∆u;
µ0 = −√me
2 e
1
3N
n∑k=1
ukσeff k
fk+1 − fk−1
21.
On the numerical code the solution corresponding to the smaller value of αR (minus sign) is used.
The larger root was also tested but either the convergence wasn’t reached or the EEDF solution was not
physically relevant (yielding negative EEDF’s values).1 At initial stages of the numerical code implementation, the calculation of the electron mobility in Argon was sensible to
variations of the energy step . For some ∆u values, the mobility was underestimated, leading to negative or complex values of
α and preventing convergence. We think that this was caused by a wrong user defined energy step. The elastic (or momentum
transfer) cross section of noble gases has a minimum (between 0.1 eV and 1eV) due to the Ramsauer-Townsend effect. This
minimum affects the mobility calculation through the effective cross section σeff . If the energy step is not small enough to
describe this minimum, the calculation of the derivative with finite differences may not be good enough.
45
With the spatial growth model a mixing of solutions for the convergence parameter αR was used.
The new αxR is calculated with equation 3.3. Then this solution is mixed with the ionization coefficient,
α0R, obtained in the previous cycle according to
αR = x · αxR + (1− x) · α0R.
The new αR is used to update the EBE matrix terms, and a new EEDF is calculated, resulting in new
CI , D and µ0 estimates, which lead to the next calculation of αxR and αR. The cycle goes on until
convergence. The calculation of the ionization coefficient through equation 3.3 is delicate, since big
changes on the EEDF may lead to negative αxR values that jeopardize the convergence. In order to
ensure a slow convergence, a small mixing parameter should be used. Usually the mixing parameter for
this growth model is x = 0.5.
The convergence tests for the spatial growth model are very similar to the temporal growth’s. The
tests are performed after the calculation of the new αR and before updating the EBE matrix. Here αR is
the convergence parameter, hence the first convergence test is
||αR − α0R||
α0R
< 10−12.
A second test is performed upon the EEDF corresponding values of two consecutive EEDFs calculating
the maximum relative differences between
max
(||fnew − fold||
fold
)< 10−10.
3.4 Numerical verification of the conservation equations
The numerical code LoKI performs a numerical verification of the conservation equations, using the
discretized form of the EBE, to control the quality of the calculations. In particular,
• the calculation of the particle balance equation is used to identify possible discretization errors;
• the calculation of the power balance equation is used to monitor the error in solving the EBE and
obtaining the EEDF.
As mentioned before (see sec 2.5.2 and 2.5.3), the particle and the energy balance equations are
obtained by multiplying the discretized form of the isotropic EBE 2.32, by 1 and u, respectively, and by
integrating the resulting matrix equation over all energy intervals. Taking, as example, the EBE for the
HF temporal growth case, one has for the particle balance and the energy balance equations
∑k
γk
−u CI√2eume
f(u)+∂
∂u
(u
3
E21R
2
σIeff (u)
σIeff (u)2
+ ω2R/
2eume
∂f(u)
∂u
)+I0(u)
γ+Jrot(u)
γ+J0(u)
γ+JI(u)
γ
√uk∆u = 0
∑k
γk
−u CI√2eume
f(u)+∂
∂u
(u
3
E21R
2
σIeff (u)
σIeff (u)2
+ ω2R/
2eume
∂f(u)
∂u
)+I0(u)
γ+Jrot(u)
γ+J0(u)
γ+JI(u)
γ
(uk)3/2∆u = 0.
46
3.4.1 Particle balance
The calculations are detailed in Annex B. Here we will only show the results for each term of the discrete
electron Boltzmann equation.
After calculations we have the following contributions:
• The exponential temporal growth term =-√
m2eCI ;
• The exponential spacial growth terms =-√
m2eCI ;
• The continuous terms = 0;
• The conservative inelastic collision terms = 0;
• The ionization collisional operator = 〈νI(u)〉N
√m2e +O(∆u).
The ionization collisional operator introduces an error of the order of the energy step, given by
O(∆u) =
(n−MI)2∑k=1
u2k+MIqisec(u2k+MI
, uk)f2k+MI∆u2,
The particle balance equation can be written, for either growth model, as
−√m
2eCI +
〈νI(u)〉N
√m
2e+O(∆u) ≈ 0.
This equation confirms that the discretized EBE preserves particle conservation, if we ignore the term
of the order of the energy grid-step.
3.4.2 Energy balance
The calculations are detailed in Annex C. Here we will only show the results for each term of the discrete
EBE.
After calculations we have the following contributions:
• The exponential temporal growth term = −√
m2eCI 〈u〉 ;
• The exponential spacial growth term = −αR
ne
√me
2 e ΓE ;
• The continuous terms =∑∞k=1(Ak −Bk)fk∆u;
• The conservative inelastic/superelastic collisions terms =√
me
2 e
∑i,jmij
(δj〈νji〉N − δi 〈νij〉N
)∆u;
• The ionization collisional operator =(MI + 1
2
)∆u√
me
2 e〈νI〉N +O(∆u).
Again, the ionization collisional operator introduces an error of the order of ∆u,
O(∆u) =
(n−MI)2∑k=1
uku2k+MIqisec(u2k+MI
, uk)f2k+MI∆u2,
47
In principle, the power lost in ionizing collisions (PI ), should be given by an similar to those of the other
inelastic collisions. However, due to the discretization of the ionization collisional operator, the correct
expression for the power lost in ionizations is
PI =
(MI +
1
2
)∆u
√me
2 e
〈νI〉N
.
in which an extra shift of ∆u/2 is introduced.
The final expression for the case of exponential temporal growth of the electron density is,
CI 〈u〉+n∑k=1
(Ak −Bk) fk∆u2
√2e
m=∑i,j
(δj〈νji〉N− δi〈νij〉N
)mij∆u+
〈νI(u)〉N
(MI +
1
2
)∆u+O(∆u) = 0,
and for the case of the exponential spatial growth,
αRne
ΓE+
n∑k=1
(Ak −Bk) fk∆u2
√2e
m=∑i,j
(δj〈νji〉N− δi〈νij〉N
)mij∆u+
〈νI(u)〉N
(MI +
1
2
)∆u+O(∆u) = 0.
48
writing of EBE matrix
w/o secondary electrons
start
calculation of initial ionization
rate coefficient (CI) Townsend coefficient (α)
update of EBE matrix with secondary electrons
calculation of EEDF
update CI or α
calculation ofconvergence
criteria
reading cross-section
data
calculation of the initial
EEDF
calculation ofpower balance
equations
output
end
end/continuecycle
choice of electron-
-density growthmodel
EBE solver
Ionization routine
Figure 3.1: Flowchart of the EBE solver in LoKI. In blue we present the original routine if secondary
electrons were not included. On the right hand side, we present all the various steps of the ionization
routine.
49
startcalculation of
calculation of EEDF with secondary
electrons and w/oe-e coll op
C I /α
test convergence
eedf and C I /α
calculation of EEDF w/o
non-linear terms
calculation of
C I /α
calculation of EEDF with secondaryelectrons and
e-e coll op
calculation of e-e coll op
test convergence
eedf
calculation of C I /α
calculation of
C I /α
test convergence of
C I /αend
test convergence
eedf and C I /α
No
No
No
No
Yes
Yes
Yes
Yes
calculation of EEDF with secondaryelectrons and
e-e coll op
Figure 3.2: Flowchart of the coupling between the ionization and the electron-electron collisions routine.
In red are the steps done within the ionization routine. In green are the steps done within the electron-
electron collisions routine.
50
Chapter 4
Results
The inclusion of secondary electrons caused a significant change on the EEDF shape for high E/N
fields. Also, the choice of the electron density growth model had a significant impact on the high-energy
part of the distribution, the so-called tail of the EEDF. Different energy sharing assumptions change the
amount of high and low energy electrons.
In order to validate the new ionization routine, it was necessary to perform benchmark tests with an
established electron Boltzmann equation solver, such as BOLSIG+ [16]. This computational tool has the
option to include secondary electron due to electron-impact ionizations with also two different electron
density growth models, exponential spatial or temporal electron density growth. However, it doesn’t
provide the option to use an energy sharing using a single differential ionization cross section (SDCS).
Consequently, these benchmarks were performed for the equal energy sharing and "one electron takes
all" cases, available in BOLSIG+.
Since the benchmark tests for a energy sharing using a SDCS could not be performed (against
another EBE solver), validation tests with experimental data were done in this case.
4.1 Comparison between energy sharing modes
Energy sharing modes refer to the way in which the scattered and secondary electrons share the energy
available after the ionization collision. Different energy sharing definitions will impact the EEDF shape in
different forms.
Two limiting cases would be the equal energy sharing, and "one electron takes all" sharing. With
equal sharing the secondary electron has always the maximum energy possible and the scattered elec-
tron the least energy possible. With the "one electron takes all" the secondary electron has always the
minimum possible energy and the scattered electron the maximum energy possible. In-between these
cases, there is the energy sharing using an SDCS.
The shape of this cross section depends on the energy of the primary electron and on the gas in
which the ionization occurs, however, there are some common features. Experimental data suggest that
there is a maximum near the zero energy of secondary electrons [32, 24], with the lowest probability
51
(eV)secu10210
/eV
2
cm
-20
10
× se
c
i q
10
210
310
2N = 100 eV∈ = 200 eV∈ = 300 eV∈
Figure 4.1: Experimental data on the differential ionization cross section on the secondary electron
energy for molecular nitrogen, and for three different primary electron energies [1].
values being for secondary electrons with energies near its maximum, that is usec ≈ (ε− VI) /2. An
example of SDCS is represented in Figure 4.1 for molecular nitrogen. According to this information, the
equal energy sharing scenario would be the least probable one, and the "one electron takes all" the
most probable one.
These tests were performed for Argon, the general behaviour being similar for other gases. A com-
parison of the three energy sharing models was done for E/N = 1000 Td 1. The different energy sharing
effects were similar between growth models, so the comparisons were done for the exponential spatial
growth model only.
With wither spatial or temporal growth, and for both energy sharing modes, there are less high en-
ergy electrons when secondary electrons are included. This is caused mostly by the introduction of the
secondary electrons at low energies that, by imposing the normalization condition, force the EEDF to be
lower at high energy values. Physically speaking, the increase in the number of low energy electrons is
high enough so that the probability of having a high energy electron is reduced. This is well illustrated in
Figure 4.2 with the energy sharing mode "one electrons takes all". This mode is in part identical to the
treatment of electron-impact ionizations as conservative inelastic collisions (see 2.3.3), the description
previously implemented in LoKI, the difference being that here secondary electrons are introduced at
zero energy. As a result, the change seems to be caused by the introduction of secondary electrons,
and not by the energy partitioning between the electrons produced after the ionization.
1For E/N = 1000Td the near-isotropy assumption, necessary to justify the two-term approximation, is not plausible since
the energy that electrons gain by the electric field during the relaxation frequency of the first anisotropy starts to be comparable
to the electron thermal velocity. However, as will be showed in sec 4.4, LoKI’s predictions are still good when compared with
experimental data. Also, higher electric fields augment the differences between energy sharing modes and electron density
growth models, facilitating this analysis.
52
u (eV)0 100 200 300 400 500 600
)
-3
/2f (
ev
20−10
18−10
16−10
14−10
12−10
10−10
8−10
6−10
4−10
2−10 Ar E/N=1000Tdwithout secondary electronsequal energy sharingone electron takes allusing a SDCS
(a) Logarithm plot of the various EEDFs for different en-
ergy sharing models.
u (eV)0 5 10 15 20 25 30 35
)
-3
/2f (
ev
0
0.005
0.01
0.015
0.02
0.025
0.03 Ar E/N=1000Tdwithout secondary electronsequal energy sharingone electron takes allusing a SDCS
(b) Linear plot of the low-energy part of the various
EEDFs for different energy sharing modes.
Figure 4.2: Plot of EEDFs calculated in LoKI for Argon with DC E/N = 1000Td and the electron density
spatial growth model.
In Figure 4.2a we can better see the behaviour of the high-energy region of the EEDF, mainly corre-
lated with the most energetic of the product electrons, the scattered electron.
In the "one takes all" case, the energy of the scattered electrons after the collision has the maximum
value usca = ε − VI . In this case the tail of the EEDF is higher, because the scattered electrons loose
less energy. Most of the high energy electrons of the distribution function will remain within the high
energy region, even after an ionizing collision.
In contrast, on the equal energy sharing case, the energy of a scattered electron after the collision is
usca = (ε− VI) /2 which is the lowest possible value for a scattered electron. This shows why this case
will have the lowest probability of finding a high energy electron. For most of these electrons, the energy
lost is high enough, to remove them from the tail into the body of the EEDF.
The tail of the EEDF for the energy sharing using a differential ionization cross section is between
these two limiting cases.
Observing Figure 4.2b we can better see the behaviour of the low energy region of the EEDF, mainly
related with the energy of the least energetic of the product electrons, the secondary electron.
For the "one electron takes all" case, the probability is the highest. This is expected, since with this
type of sharing secondary electrons are introduced at zero energy usec = 0.
The EEDF of the equal energy sharing case has the lowest probability (apart from the case without
secondary electrons). This can be explained by the fact that with this type of energy sharing, secondary
electrons have the highest possible energy usec = (ε− VI) /2.
Again, the case in which the energy sharing is described with a differential ionization cross section
is between the previous limiting cases.
53
It is interesting to see that the EEDF for the equal energy sharing case has a probability higher than
the other distributions between 7eV and 20eV. This explains why it has the lowest probability both for
low and high energy regions, while complying with the EEDF normalization condition. An analogous
observation can be made for the "one electron takes all" energy sharing mode.
4.2 Comparison between electron density growth models
Electron-density growth models refers to the way in which the introduction of secondary electrons influ-
ence the electron density. In this work, two electron density growth models were adopted, assuming a
spatial growth and a temporal growth.
The results obtained using these models are shown in Figure 4.3, for a DC Argon plasma at E/N =
1000Td, where we have also included the case in which secondary electrons are not considered.
u (eV)0 100 200 300 400 500 600
)
-3
/2f (
ev
19−10
17−10
15−10
13−10
11−10
9−10
7−10
5−10
3−10
Ar E/N=1000Td without secondary electronsexponential temporal growthexponential spatial growth
(a) Logarithm plot of the various EEDFs for different elec-
tron density growth models.
u (eV)0 5 10 15 20 25 30 35
)
-3
/2f (
ev
0
0.005
0.01
0.015
0.02
0.025
0.03Ar E/N=1000Td
without secondary electronsexponential temporal growthexponential spatial growth
(b) Linear plot of the low-energy part of the various
EEDFs for different electron density growth models.
Figure 4.3: Plot of EEDFs calculated in LoKI for Argon with DC E/N = 1000Td and the energy sharing
using a SDCS.
A significant difference is observed between the two growth models. Spatial growth seems to pro-
duced lower tails than the temporal growth model, which also relates to a higher probability of finding
low energy electrons.
This result is coherent with lower drift velocities for the spatial growth model, where the electron flux
is composed by two opposite components, due to the drift in the electric field and the diffusion caused
by a pressure gradient.
electron density exponential spatial growth ~Γ = −∇r(Dne) + µ0ne ~E;
electron density exponential temporal growth ~Γ = µ0ne ~E.
54
4.3 Benchmarks against BOLSIG+
One of the most popular EBE solvers is BOLSIG+ [16]. This EBE solver has some of the electron density
growth models of LoKI. However, it can only use either the equal sharing or the "one electron takes all"
energy sharing mode. Thus, our benchmark tests against BOLSIG+ will not include the description of
ionization using a differential cross section.
The first tests were for the EEDF in Argon at E/N = 1000Td, with the temporal growth and the spatial
growth models. After, the first Townsend ionization coefficient, calculated with LoKI and BOLSIG+, was
compared for different E/N values in Argon and in molecular Nitrogen.
u (eV)0 100 200 300 400 500 600
)
-3
/2f (
ev
11−10
10−10
9−10
8−10
7−10
6−10
5−10
4−10
3−10
2−10BOLSIG equal energy sharing
LoKi equal energy sharing
BOLSIG one takes all energy sharing
LoKi one takes all energy sharing
Ar E/N=1000Td electron density exponential temporal growth
(a) Comparisons between LoKI and BOLSIG+ for differ-
ent energy sharing in ionization, using the exponential
temporal growth model for the electron density.
u (eV)0 100 200 300 400 500
)
-3
/2f (
ev
16−10
15−10
14−10
13−10
12−10
11−10
10−10
9−10
8−10
7−10
6−10
5−10
4−10
3−10
2−10 BOLSIG equal energy sharing
LoKi equal energy sharing
BOLSIG one takes all energy sharing
LoKi one takes all energy sharing
Ar E/N=1000Td electron density exponential spatial growth
(b) Comparisons between LoKI and BOLSIG+ for differ-
ent energy sharing in ionization, using the exponential
spatial growth model for the electron density
Figure 4.4: Comparisons of the EEDF, calculated with LoKI and BOLSIG+, for Argon with DC E/N =
1000Td.
First we will compare EEDFs calculated with BOLSIG+ and LoKI for equivalent electron-density
growth models and energy sharing modes. Contrary to LoKI, BOLSIG’s energy grid is not completely
defined by the user. BOLSIG+ discretizes the electron density growth and electric field terms using an
exponential scheme [33], which is accurate for some convection and diffusion conditions [16], meaning
that it automatically defines the number of energy grid-points and energy limit for a given set of dis-
charge conditions. Comparing two distribution functions with different initial and final energy grid points
is difficult, since different energy limits can change the EEDF in a significant way when enforcing the
normalization condition. So, whenever possible, we have tried to match LoKI’s user-defined energy grid
with BOLSIG’s.
For the exponential temporal growth model there is very good agreement between BOLSIG+ and
LoKI for all energy sharing cases.
The EEDFs for the exponential spatial growth model are similar in shape, and in the trend observed
55
for the different energy sharing modes, but they don’t match as well as in the temporal case. One of the
factors for this discrepancy was the difficulty in matching the energy grid-points. If the BOLSIG’s upper
limit for the energy grid is used in LoKI, large power-balance errors are present. However, even when
these limits are made very similar, there are still some differences between the EEDFs that might be
due to the distinct discretization procedures in the two EBE solvers.
E/N (Td)210
310
2
/N m
α
0
5
10
15
20
25
30
35
40
45
21−10×Ar DC discharge
BOLSIG+ equal energy sharing
LoKI equal energy sharing
BOLSIG+ one electron takes all sharing
LoKI one electron takes all sharing
(a) First Townsend ionization coefficient in Argon DC
plasmas for various E/N values.
E/N (Td)210
310
2
/N m
α
0
5
10
15
20
25
30
35
21−10× DC discharge2N
BOLSIG+ equal energy sharing
LoKI equal energy sharing
BOLSIG+ one electron takes all sharing
LoKI one electron takes all sharing
(b) First Townsend ionization coefficient in Nitrogen DC
plasmas for various E/N values.
Figure 4.5: Comparison between first Townsend ionization coefficient calculated with BOLSIG+ and
LoKI, using the exponential spatial growth model and adopting different energy sharing modes.
For the first Townsend ionization coefficient, the EBE solvers predict similar results. However, there
are some deviations for high E/N values in Nitrogen and even for low E/N values in Argon.
There is a strange behaviour, more evident in nitrogen, around E/N = 700Td, where BOLSIG’s
predictions deviate, when compared with LoKI’s prediction path. Between E/N = 1000Td and E/N =
2000Td, BOLSIG’s predictions for the "one electrons takes all" case approach LoKI’s corresponding
energy sharing mode prediction.
4.4 Validation of the first Townsend ionization coefficient against
experimental data for Ar and N2
In order to assess the improvements in the values of transport coefficients predicted by LoKI, a com-
parison of the first Townsend ionization coefficient with experimental data was made. The experiments
were made in a Steady State Townsend (SST) discharge, accordingly the spatial electron-density growth
model was adopted. Comparisons will use the energy sharing using an SDCS and the case in which
secondary electrons are not included.
In Argon the simulation predictions improved significantly but they are still above experimental data
56
E/N (Td)210
310
2
/N m
α
21−10
20−10
19−10
Argon SST discharge
Exp data
LoKI with secondary electronsLoKI without secondary electrons
(a) Comparison between LoKI’s calculated first
Townsend ionization coefficient and experimental data
for Argon SST discharges [34].
E/N (Td)210
310
2/N
mα
24−10
23−10
22−10
21−10
20−10
19−10 SST discharge2N
Exp dataLoKI with secondary electronsLoKI without secondary electrons
(b) Comparison between LoKI’s calculated first
Townsend ionization coefficient and experimental data
for Nitrogen SST discharges [2].
Figure 4.6: First Townsend ionization coefficient as a function of the reduced electric field. LoKI’s simula-
tions use the exponential spatial growth model or conservative ionization collisions (secondary electrons
not included).
uncertainty. In Nitrogen the inclusion of secondary electrons was enough to shift LoKI’s predictions into
experimental data uncertainty. These results show that LoKI can now operate at higher E/N values.
LoKI does yet not consider attachment or recombination non-conservative processes. Since attach-
ment and recombination have an effect on the number of electrons, it may have an influence on the
electron density growth terms of the EBE (in the case of Argon and molecular Nitrogen, attachment
would not improve prediction since these gases are not electronegative).
In a discharge there are other ionization processes that contribute to the spatial growth of the elec-
tron density, such as associative and Penning ionization, chemionization, photoionization, associative
detachment, and ionization by neutral particles and ions [35, Ch. 9.3]. Photoionization and heavy-
particle collisional ionizations are not included in this EBE, but they do influence the EEDF.
In SST experiments, the ejection of electrons from the cathode due to ions or photons is significant.
These secondary processes can be difficult to separate from the current growth due to electron impact
ionization. To avoid this problem, some experiments are made in a time scale shorter than the ions
transient time [35, Ch. 9.3]. On the Argon SST experiment [34], the first Townsend coefficient was
measured through the spatial variation of light emitted by the de-excitation of metastable states. This
spatial measurement was done by moving the detector in a platform with a stepper motor controlled by
a computer, with the light ouptup being measured in 1mm steps until 25mm [34]. Although there is no
reference to the total time that these measurements took, it is safe to assume that these measurements
were not shorter than the ions transient time, or the metastables diffusion time, and so, electrons from
57
E/N (Td)3
10
)2/N
(m
α
20−10
SST discharge2NExp dataequal energy sharingone electron takes all
Figure 4.7: Comparison between LoKI’s calculated first Townsend ionization coefficient with equal en-
ergy sharing mode, "one electron takes all" mode, and experimental data for Nitrogen SST discharges
[2].
ion and photon cathode emissions probably played a role on the electron density spatial profile. In this
experiment, the first Townsend coefficient α was calculated by adjusting an exponential spatial growth
of the electron density, similar to 2.19, to the measured electron density profile. In sum, the measured
first Townsend ionization coefficient, was done by measuring an electron density spatial profile created
by electron-impact ionization and possibly by other processes that are not included in the EBE solver.
Some measurements of the secondary ionization coefficients were made in nitrogen [36], and sep-
arated in fast and slow coefficients, due to cathode emissions by ions and by photons or metastables,
respectively. With this discrimination of the secondary ionization coefficients, it may be possible to in-
clude these processes in the EBE.
4.4.1 The use of the equal energy sharing mode
The non-conservative ionization operator with predefined equal energy sharing is popular in the LTP
community. One of the reasons is that the numerical writing of this operator uses much less computa-
tional resources compared to the one with energy sharing using a SDCS. However, the equal energy
sharing mode continues to be preferred even when the "one electron takes all" mode is available, and
despite the fact that the latter consumes comparable resources. This may be because it leads to closer
estimates when compared to the experimental data of the first Townsend ionization coefficient (see
Figure 4.7).
Nonetheless, we have seen in section 4.1 that the equal energy sharing scenario is much less prob-
able than the "one electron takes all" scenario (when compared to a differential description of the energy
sharing), so the better estimates cannot be attributed to a more realistic ionization operator. Also in
section 4.1, we have seen that the EEDF can suffer significant changes between energy sharing modes.
58
One of the possibilities, is that the EEDF calculated with the equal energy sharing mode, somehow
underestimates the value of the first Townsend ionization coefficient, leading to closer estimates.
Although the equal energy sharing mode leads to better estimates of the first Townsend ionization
coefficient, the fact that it is the least probable scenario means that its use cannot be properly justified
with physical arguments. A better option to improve the calculation of the first Townsend ionization coef-
ficients would be to include the other non-conservative processes that were discussed in this section.
59
60
Chapter 5
Prospective
An ionization routine was fully integrated into LoKI-B. Both the ionization routine, and electron-electron
collisions coupling strategies are outlined in work-flows, aiding in future changes. This work also pro-
vides a general description of LoKI’s EBE solver operating procedure.
The basic functionalities of the ionization routine have been developed. It has opened some paths
to calculate new coefficients and include some more mechanisms. However, some numerical improve-
ments and tests can be made as well in order to help users, as well as other validations with experimental
data.
• Now that electron density growth is included on the EBE, the path to include other non-conservative
processes is simplified.
Attachment collisions can easily be included with a collisional operator. Recombinations can also
be included but they are more complex, due to it being a three body process.
For some gases, such as nitrogen, frequencies of electron cathode ejection for fast processes
(photoionization and ion impact ionization) and for slow processes (metastable impact ionization)
have been measured. This approach is not common, but may improve the similarities between the
measured and simulated first Townsend coefficient;
• The spatial growth model permits the calculation of new parameters, such as the longitudinal
diffusion coefficient and the bulk drift velocity [17]. It is also useful when doing electron density
gradient expansions. The implementation of these calculations would allow to take full part of the
spatial growth model;
• Inclusion of a nonlinear standard algorithm in the ionization routine, such as Newton-Raphson or
Broyden’s method.
In some situations, of very high/low electric-field, very large number of grid-points, and high ioniza-
tion degree, the convergence can be quite difficult. These methods could diminish the number of
iterations required for convergence, as well as improve the convergence pathway when including
electron-electron collisions;
61
• In many cases, LoKI performs sequential batch simulations, e.g. for increasing values of the elec-
tric field, in which case many operating conditions remain the same (e.g. the number of intervals
and the maximum value of the energy grid). In these cases, it is not necessary to re-write the
ionization collisional operator for each simulation, which could significantly reduce run times;
62
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