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Modelling of Reduced Electric Field and Concentration Influence on Electron Transport Coefficients of He-Ne Plasma

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ACTA PHYSICA POLONICA A No. 4 Vol. 140 (2021)
Proceedings of the Mustansiriyah International Conference on Applied Physics (MICAP-2021)
Modelling of Reduced Electric Field and
Concentration Influence on Electron Transport
Coefficients of He–Ne Plasma
Maysam T. Al-Obaidi, Rafid A. Aliand Baidaa Hamed
Department of Physics, College of Science, Mustansiriyah University, Baghdad, Iraq
Doi: 10.12693/APhysPolA.140.299 e-mail: rafidphy_1972@uomustansiriyah.edu.iq
A comprehensive theoretical investigation on the specific electron transport coefficient in the He–Ne
gas mixtures state in plasma has been carried out. The performed calculations enable us to measure
the influence of the concentration of He:Ne gas mixture on plasma electronic coefficients. Our approach
is based on the variation in the resistance of plasma field placed in a varying reduced electric field
and thermodynamic equilibrium. The effects of reduced electric field and mixture concentrations on the
electron mobility, diffusion coefficients, and total collision frequency have been calculated by solving the
Boltzmann equation using the two-term approximation. The Monte Carlo simulation was used to solve
the Boltzmann equation. The polynomial and logistic functions were adopted through the simulation
process, and the appropriate equations have been provided, indicating the variation of plasma electronic
coefficients to the variation of gas mixture concentration and reduced electric field. The applied reduced
electric field has been chosen to be in the limited range of 0–100 Td, and is considered for several
concentrations in the limited range of 0.1–0.7 mol. The study shows that the obtained results are in
good agreement with the results obtained using the Monte Carlo method.
topics: gas discharge plasma, electron transport coefficients, electron energy distribution function
1. Introduction
Plasma is defined as a “moldable” material [1] and
is one among the four elemental states of matter,
which was first expounded by chemist Irving Lang-
muir during the 1920s [2]. The term “plasma” is
typically coined to explain a broad range of macro-
scopic neutral matters consisting of many closely
interacting and colliding free electrons and ionized
atoms or molecules, which show amalgamated be-
haviors and properties owing to the presence of
long-range Coulomb energies and forces. It should
be noted that the entire charged substances or par-
ticles field could not be brought under the clas-
sification of plasmas. There are some conditions
or criteria to be satisfied to classify charged sub-
stances as plasma and to exhibit the plasma proper-
ties [3]. For instance, an ionized substance, which is
macroscopically neutral with a characteristic length
smaller than the Debye length could be considered
as plasma. The primary interactions in plasma are
electromagnetic interactions that essentially origi-
nated from the presence of an abundant number of
charged substances/particles in plasma [4, 5].
Since plasma largely contains electrically con-
ductive gaseous ionized substances, it enables the
long-range electric and magnetic fields to con-
trol the behavior and properties of matters. The
characteristics such as the existence of high conduc-
tivity and response to the electric/magnetic fields
make plasma potentially useful in a wide variety of
applications. Firstly, a suitable control is needed.
Secondly, the potential energy sources or radiations
are required to achieve the plasma. Therefore, the
lasers are routinely harnessed to produce a variety
of plasma types in the laboratory [6, 7].
Furthermore, the characteristic properties of
plasma are remarkably dependent on the interac-
tions among the substances in plasma. Notably,
the presence of “collective effects” in plasma is
often found to be one of the fundamental fea-
tures that differentiate the plasma from that of
the other typical solids and fluids. Furthermore,
it is because of the existence of long-range elec-
tromagnetic forces. Every individual charged sub-
stance in the plasma simultaneously interacts with
more number of other charged particles, manifest-
ing the “collective effects” which are the key fac-
tors for the effective physical phenomena occurring
in plasma [8, 9].
The precise collision cross-section and rate co-
efficient values are crucial for determining the ki-
netic processes through modeling [1–3, 8, 9], simu-
lation [4] and diagnostics [5, 6, 8, 9] of the discharge
processes [10, 11].
299
Maysam T. Al-Obaidi et al.
It is known that the plasma at low tempera-
ture contains an adequate amount of neutral sub-
stances, where their greater internal degree of
freedom enables their active collisions and interac-
tions with electrons in the plasma. The existence
of surplus electrons and other interaction chan-
nels, such as ionization/recombination reactions for
heavy plasma substances, substantially leads to the
reformulation of electron transport equations and
mathematical expressions for various factors, such
as electron diffusion, electrical conductivity, ther-
mal conductivity, and viscosity coefficients. Thus,
the role of inelastic collisions of electrons towards
the deduction of electron transport coefficient has
become the subject of interest [12, 13].
The electron transport processes in plasma and
gaseous substances have been extensively investi-
gated over the last few decades through theoretical
and experimental studies. Most of the theoretical
investigations were devoted primarily to the analy-
sis of scalar transport features of electrons since the
rate of chemical reactions and electron energy re-
laxations are crucial and constitute the focal point
of research in plasma [14].
The fluid model-based gas discharge processes
typically require the transport- and rate-coefficients
as the input. The coefficients are typically estimated
from the collision cross-section data by using the
electron energy Boltzmann equation (EEBE) [15],
and are largely associated with the electron en-
ergy distribution function (EEDF). In fact, the
Maxwell–Boltzmann distribution functions which
are obtained from the kinetic theory of gaseous sub-
stances provide a facile description for many basic
gas properties including the properties such as dif-
fusion, pressure, and velocity [16, 17].
In the present work, a mathematical model which
is devoted to theoretical investigation on specific
electron transport coefficient of He–Ne plasma is
predicted by solving Boltzmann equation utilizing
the two-term approximation and by taking the ad-
vantages of Monte Carlo simulation method.
2. Mathematical model
In this work, the BOLSIG+program has been
employed to solve the Boltzmann equation (BE)
using the direct simulation Monte Carlo (DSMC)
method which is based on the Monte Carlo (MC)
simulation for the finite number fluid-flows [18].
BOLSIG+is considered a more generic and easier-
to-use tool as compared to other BE solving meth-
ods. A wide variety of fundamental processes in
gas–discharging plasma can be assessed and esti-
mated using BE which typically include the process
of electron–atom collisions, elastic collisions, elec-
tron impacted atom excitation, the transition of col-
lisions between the surrounded atoms, and the ion–
atom collisions resulting from the elastic collisions.
In the present work, the electron mobility, diffusion
coefficients, and total collision frequency have been
calculated with varying reduced electric field E/N,
several concentrations of helium–neon (He–Ne) gas
mixture, above the excitation temperature of 300 K.
Variations of mixture concentration can specify
the He–Ne plasma electronic transport coefficients
for many applications.
2.1. The Boltzmann equation
Boltzmann introduced the probability via ex-
pounding the statistical characteristics of a non-
equilibrium thermodynamic system. Probability
has also been used in quantum physics to describe
the concept of time reversal, associating the direc-
tion of probability with the entropy exchange and
breaking the similarity between the past and the fu-
ture [19, 20] — even though the laws of mechanics
remain unchanged for time inversion [21, 22].
The Boltzmann relation gives the ratio of number
density (i.e., number per unit volume) of the sub-
stances such as atoms, ions or molecules at a certain
energy level N2to the number density in lower en-
ergy level N1. This can be written as [23]
N2
N1
=G2
G1
exp E
kBT,(1)
where G1and G2denote the multiplicity of two en-
ergy positions, Eis the energy required to excite
the particles, kBis the Boltzmann constant, and T
is the temperature. Notably, a greater number of
particles can be excited by increasing T[18, 20].
The Boltzmann equation (BE) is
∂f
∂t =f
∂t force
+∂y
∂x diff
+∂f
∂t coll
,(2)
where the first term represents “force” and it can
be influenced by the external forces acting on the
particles. In turn, the second term “diff indicates
the particle diffusions. Finally, the term “coll” in-
dicates the forces acting between the particles in
collision [24, 25].
2.2. Equations of force, diffusion
and collision terms
The force term and diffusion term corresponding
to (2) can be written, respectively, as
∂f
∂t force
=F(r, t)·∂f
p(3)
and
∂y
∂x diff
=p
m· ∇f. (4)
The equivalent form of (2) is then
∂f
∂t =p
m· ∇f+F(r, t)·∂f
p+∂f
∂t coll
,(5)
Here, F(r, t)represents the field exerting on the
particles in fluid, pis the momentum, and mis
the particle mass. The terms on the right side of
(5) describe the effects of particle collisions, where
it will be zero if there is no collision between the
particles [26]. Regarding the “collision” term, many
300
Modelling of Reduced Electric Field and Concentration. . .
efforts have been taken towards modelling the col-
lision. In this direction, the models/equations pro-
posed by Bhatnagar, Gross and Krook (known as
BGK form) [26] is one of the well-known mod-
els/equations. The modified BE as per the BGK
form is given as
∂f
∂t +p
m· ∇f+F·∂f
p=ν(f0f),(6)
where νdenotes the frequency of collision and f0is
the local Maxwillian distribution function.
3. General Boltzmann equation
for a particle mixture
For a chemical mixture of particles categorized
by the indices i= 1,2,3, . . . , n, one has the
following
∂fi
∂t +pi
mi
· ∇fi+F(r, t)·∂fi
pi
=∂fi
∂t (7)
with fi=fi(r,pi, t)and where rrepresents the
position of particle, piand miare the momentum
and mass of species, respectively. The collision
term can be given by
∂fi
∂t coll
=(8)
n
X
j=1 ZZ d3p0dgij Iij (gij )f0
if0
jfifj.
It can be seen that the integration is over the
components of momentum, where f0=f0(p0
i, t),
Iij represents the differential cross-sections between
the species iand j, and ddenotes the solid an-
gle of collisions. The relative momentum gij can be
given by
gij =|pipj|=
p0
ip0
j
,(9)
where pi,pjand p0
i,p0
Jare the i, j-th momentums
before and after collisions, respectively [27].
4. Results and discussion
Figure 1a and b represents the collisional cross-
section area of He and Ne. To solve the BE,
one needs substantial information about the colli-
sional cross-section of the gas mixture considered.
Figure 1 shows the cross-section with respect to the
electron energy of the reaction involved. Difference
in atomic structure between He and Ne leads to dif-
ferent energy levels for each gas, and consequently,
to theirs different behavior (as can be seen in Fig. 1a
and b). The output data has been included in the
current calculations.
Figure 2a and b shows that there is a significant
dependence of the electron energy distribution func-
tion (EEDF) on the electron mean energy (for sev-
eral values of applied electric field E/N). The num-
ber of particles is N= 2.7×1025 m3. The obtained
results also show that the electrons can be affected
by a forward bias by increasing the applied elec-
tric field E/N. This bias will accelerate the elec-
trons and increase energy up to a steady level of
Fig. 1. The collisional cross-section of (a) He and
(b) Ne as a function of electron energy.
Fig. 2. EEDF of (a) He and (b) Ne as a function
of the electron energy.
301
Maysam T. Al-Obaidi et al.
mean energy, which eventually increases the rate of
inelastic collisions. Above this point of the steady
energy level, the increment in E/N will be ineffec-
tive and the curve will fall towards the high-energy
tail. In other words, the high values of EEDF will
be at the low values of energy u. When increasing
the uvalues, the EEDF will approach the original
point due to the high energy electron resulting from
the applied forward bias. Values of EEDF for several
concentrations of He–Ne mixture are quasi logarith-
mic according to the slope of 1/(kBT). The shape
of the curves, as displayed in Fig. 2, illustrates the
Maxwell distribution for the several mixture con-
centrations.
4.1. Influence of E/N on the electron mobility
In Fig. 3, the variation in electron mobility µ/N
can be observed with respect to the increasing E/N
for several concentrations (0.1–0.6 mol) of He–Ne
mixtures, where the sharp decrease of µ/N occurs
between 0 and 40 Td. In the further range of E/N,
i.e., 40–100 Td, the mobility values become almost
steady. This can be caused by the descending elec-
tron energy as a result of ionization and excitation
processes [28].
Fig. 3. Mobility as a function of different concen-
trations.
Fig. 4. Change in energy mobility with respect to
E/N (obtained by the BOLSIG+and the estimated
data).
4.2. Influence of the mixture concentration
on the mobility
In Table I, the influence of mixture concentration
on electrons mobility can be observed by analyzing
the Cbehavior from a low 0.1 mol to its higher
value 0.7 mol. The observed effect results from the
excitation and ionization processes in relation to the
reduced electric field E/N.
4.3. Modeling of the mobility
We have calculated the concentration values us-
ing the BOLSIG+program which bases on the de-
scription of the mobility behavior (µ/N ) depend-
ing on E/N. The obtained results are presented
in Fig. 4. Notably, the present BOLSIG+data are
found to be well-matched with the data obtained in
our present work (see Fig. 4).
The fitting relationships for several concentra-
tions can be written as
µ
N=A2+A1A2
1 + E/N
X0P,(10)
where E/N = 0–100 Td, while A1,A2,X0,P,
for a given concentration value, are constants. Poly-
nomial form has been assumed for
A1=3.32 + 0.9C+ 4.87C2×1024,(11)
A2=2.91 + 0.46C+ 2.34C2×1024,(12)
X0= 36.6+7.8C16.22C2,(13)
P= 2.65 0.6C+ 0.1C2,(14)
to determined the required concentration C.
4.4. Influence of E/N on the diffusion
coefficients DN
Figure 5 shows the variation in diffusion coeffi-
cient (DN) with respect to the variation in the elec-
tric field E/N. A slight/small increment in DN is
observed in the limited range of 0< E/N < 20 Td,
while a clear DN increment is observed in the range
of 20 Td<E/N = 20–60 Td. Next, in the range
of 60–100 Td, the DN values increase slightly.
TABLE I
Influence of the mixture concentration Con the elec-
tron mobility µ/N (E/N = 55.62 Td).
Concentration C[mol] µ/N ×1025 [1/(m V s)]
0.1 0.254
0.2 0.237
0.3 0.287
0.4 0.320
0.5 0.350
0.6 0.387
0.7 0.433
302
Modelling of Reduced Electric Field and Concentration. . .
Fig. 5. Diffusion coefficient as a function of E/N
at different concentration values C.
Fig. 6. Diffusion coefficient as a function of E/N
(obtained by BOLSIG+and our estimated data.
TABLE II
Influence of mixture concentration Con diffusion co-
efficient DN (E/N = 41.97 Td).
Concentration C[mol] DN ×1025 [1/(m s)]
0.1 0.44
0.2 0.48
0.3 0.53
0.4 0.58
0.5 0.65
0.6 0.74
0.7 0.85
The growth of DN is essentially due to the electron–
ion and electron–n collisions, which caused the drift
of the mass centre of atom (ion) and a subsequent
rerelease of electrons.
4.5. Influence of the concentration on DN
Table II shows the variation in the diffusion coef-
ficient DN with respect to the variation of mixture
concentration Cfor a specific value of reduced elec-
tric field E/N = 41.97 Td. This direct relationship
between DN and Cis due to the inelastic collisions.
4.6. Diffusion coefficients modeling
The behavior of DN with respect to the change
in E/N is shown in Fig. 6. The estimated points are
obtained using the logistic function of the form
DN =B2+B1B2
1 + E/N
r0r,(15)
where E/N = 0–100 Td, and B1,B2,r0,rare con-
stants for a given values of C. Therefore, the fitting
for all concentrations is reflected with (15).
The parameters B1,B2,r0,rhave been obtained
utilizing polynomial function, respectively, as
B1=2.81 + 0.892C+ 0.356C2×1024,(16)
B2=9.50.615C+ 6.9C2×1024,(17)
r0= 73.334.16C6.55C2,(18)
r= 2.36 0.073C0.3C2.(19)
Therefore, the concentration required in (15) is de-
termined with the use of (16)–(19).
Figure 6 shows a well-matched result between the
data obtained by the BOLSIG+program and the
estimated data with (15) for a specific concentration
value C= 0.5mol.
4.7. Influence of E/N on total collision frequency
Figure 7 shows the dependence of total collision
frequency (TCF) on E/N of the He–Ne mixture.
In the range of E/N < 20 Td, a slight increment
in the TCF is observed, whereas a sharp TCF in-
crement is observed for E/N > 20 Td. This result
is referred to the fact that the electric energy and
cross-section depend on the electron excitation and
ionization energy [28].
Table III shows the values of TCF of the He–Ne
mixture for a specific value of the reduced electric
field. High TCF values are observed at a low mix-
ture concentration of 0.1 mol compared to a high
mixture concentration of 0.7 mol. This can be as-
cribed to the high inelastic collisions rate and the
collisional cross-section at low concentrations.
Fig. 7. Total collision frequency as a function of
E/N at different concentrations C.
303
Maysam T. Al-Obaidi et al.
4.8. Total collision frequency (TCF) modeling
The behavior of TCF with respect to the vary-
ing values of E/N is shown in Fig. 8. The esti-
mated points of the total collision frequency νeoT
are obtained using the logistic function of the
form
νeoT
N=G2+G1G2
1 + E/N
n0n,(20)
where E/N = (0–100) Td.
The parameters G1,G2,n0,nwhich are constants
for a given value of C, are obtained using polyno-
mial function, respectively,
G1=4.25 3.22C8.38 ×104C2×1014,
(21)
G2=7.56 5.2C+ 0.176C2×1014,(22)
n0= 48.03 2.68C7.28C2,(23)
n= 2.47 0.039C0.486C2.(24)
Therefore, the concentration Crequired in (20) is
determined with the use of (21)–(24).
Fig. 8. Total collision frequency results show well-
matching between the BOLSIG+and the estimated
data under specific concentration Cof 0.5 mol.
TABLE III
Total collision frequency νeoT in the He–Ne mix-
ture under the effect of the reduced electric field
(E/N = 525.6Td).
Concentration [mol] νeoT/N ×1013 [m3/s]
0.1 0.578
0.2 0.536
0.3 0.494
0.4 0.453
0.5 0.411
0.6 0.369
0.7 0.328
5. Conclusions
The effect of induced electric field and plasma
state in the He–Ne gas mixture concentrations on
the electron energy distribution function (EEDF)
has been investigated. The results are obtained with
the Boltzmann equation by the BOLSIG+program
supported by the direct simulation Monte Carlo
method. It is observed that increasing the elec-
tric field and the concentration of mixture increase
the EEDF due to inelastic collisions which influ-
ence the behavior of EEDF and are closely related
to the electronic coefficients. The changes observed
in EEDF with respect to the electric field indi-
cated that the performed calculations and results
obtained agree with the Maxwell equations.
The logistic and polynomial functions were fit-
ted to the data. We have obtained the appropri-
ate and simplified equations to observe the behav-
ior of transport electronic coefficients. The results
also showed that the data from the BOLSIG+
Monte Carlo method were found to be well-matched
with the estimates obtained in the fitting process
performed in this study. Also, the output results
turned out to be well suited to other studies [10, 27].
Overall, the obtained results suggested that the
Boltzmann equation (BE) can be one of the most
perspective tools to investigate the plasma states
and their electronic coefficients from the kinetics of
electrons in feebly ionized gaseous substances.
Acknowledgments
Authors would like to express their thanks and
gratitude to the Department of physics, College of
Science, Mustansiriyah University for support and
cooperation.
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... The kinetic or transport equations are generalizations of the Boltzmann equation and showcase electron gas behaviors in crystal lattice phonons and in metals [23,24]. ...
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