Foundations of modelling of nonequilibrium low-temperature plasmas

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Figures

Table 2. List of physical constants.

Figure 5. Ratio of the electron effective diffusion coefficient to the ambipolar diffusion coefficient, as a function of the ratio of the characteristic diffusion length to the ion mean-free-path, according to [65,66].

Figure 6. Reduced electric field Eext/n0, axially applied to a DC helium discharge at Tg = 300 K, as a function of the product n0R (with R = 2.405Λ the discharge radius in cylindrical geometry). The curves are calculations for IDC = 50 mA (solid line) and 200 mA (dashed). The points are measurements of Kaneda [73] for IDC = 30 mA.

Figure 10. Calculated time-evolution of the gas-phase concentrations of NO (blue curves) and NO2 (red) from a KMC (—) and DD (– –) model, for various pressures. Taken from [151].

Figure 9. Simplified schematic of a surfatron microwave plasma (a), with power deposition and corresponding gas temperature profile as assumed in the model (b).

Figure 2. Electron energy distribution functions in a radio-frequency discharge in nitrogen, measured using a Langmuir probe (points) and computed from a PIC-MCC simulation (lines). This result shows the capacity of the PIC method for computing complicated energy distributions in good agreement with experiment (from [16]).

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This item is the archived peer-reviewed author-version of:
Foundations of modelling of nonequilibrium low-temperature plasmas
Reference:
Alves L.L., Bogaerts Annemie, Guerra V., Turner M.M..- Foundations of modelling of nonequilibrium low -temperature plasmas
Plasma sources science and technology / Institute of Physics [Londen] - ISSN 0963-0252 - Bristol, Iop publishing ltd, 27:2(2018), 023002
Full text (Publisher's DOI): https://doi.org/10.1088/1361-6595/AAA86D
To cite this reference: https://hdl.handle.net/10067/1493910151162165141
Institutional repository IRUA

Foundations of modelling of nonequilibrium
low-temperature plasmas
L. L. Alves
1
, A. Bogaerts
2
, V. Guerra
1
, M. M. Turner
3
1
Instituto de Plasmas e Fus˜ao Nuclear, Instituto Superior T´ecnico, Universidade de
Lisboa, Av. Rovisco Pais, 1049-001 Lisboa, Portugal
2
Department of Chemistry, Research Group PLASMANT, University of Antwerp,
Universiteitsplein 1, B-2610 Antwerp, Belgium
3
School of Physical Sciences and National Centre for Plasma Science and
Technology, Dublin City University, Dublin 9, Ireland
E-mail: llalves@tecnico.ulisboa.pt
December 21, 2017
Abstract. This work explains the need for plasma models, introduces arguments for
choosing the type of model that better fits the purpose of each study, and presents
the basics of the most common nonequilibrium low-temperature plasma models and
the information available form each one, along with an extensive list of references
for complementary in-depth reading. The paper presents the following models,
organized according to the level of multi-dimensional description of the plasma: kinetic
models, based on either a statistical particle-in-cell / Monte-Carlo approach or the
solution to the Boltzmann equation (in the latter case, special focus is given to the
description of the electron kinetics); multi-fluid models, based on the solution to the
hydrodynamic equations; global (spatially-average) models, based on the solution to
the particle and energy rate-balance equations for the main plasma species, usually
including a very complete reaction chemistry; mesoscopic models for plasma-surface
interaction, adopting either a deterministic approach or a stochastic dynamical Monte-
Carlo approach. For each plasma model, the paper puts forward the physics context,
introduces the fundamental equations, presents advantages and limitations, also from
a numerical perspective, and illustrates its application with some examples. Whenever
pertinent, the interconnection between models is also discussed, in view of multi-scale
hybrid approaches.
Keywords: low-temperature plasmas, modelling, particle-in-cell, kinetic equation, fluid
model, global model, plasma-surface mesoscopic model
Submitted to: Plasma Sources Sci. Technol.

Foundations of modelling of nonequilibrium low-temperature plasmas 2
1. Introduction
The field of low-temperature plasma (LTP) science and engineering has been driven
by both fundamental science issues and applications targeting societal benefits [1].
Fundamental research strives to gain deeper understanding of the underlying principles
governing these plasmas, and is essential to ensure the scientific advances required for
consolidated successful applications.
Modelling activities are a fundamental component of any research field, to
complement and/or assist the implementation of experimental diagnostics; to give
predictions on the behaviour of meaningful quantities, especially when these are
inaccessible experimentally; and to deepen fundamental knowledge of the field. The
modelling and simulation of LTPs has been considered a requirement for the progress
in the field [1,2], and model-based design for plasma equipment and processes has been
identified as a necessary capability to achieve industrial goals [3]. Indeed, the guidance
provided by LTP models is particularly relevant due to the extreme complexity of the
medium (often exhibiting different material phases), composed by charged particles
(electrons and ions) and by neutral species in different excited states, intrinsically in
non-equilibrium as result of collisional, radiative and electromagnetic interactions.
The success in the modelling and simulation of LTPs is impressive, given the
extreme diversity of the field, including types of plasma sources, pressures, spatial and
time scales, electron energies and chemistries. This diversity can only be met by a
suite of models, adopting formulations and algorithms adapted to the specific working
conditions and features of each gas/plasma system. This work integrates the collection
of papers published by Plasma Sources Science and Technology on Foundations of
Low-Temperature Plasmas and Their Applications, conveying essential information
to enable a beginner in this field to identify the most common nonequilibrium LTP
models and the information available from each one. Models of plasmas in local
thermodynamic equilibrium are addressed in a different paper of the same collection [4],
on the “Foundations of high-pressure thermal plasmas”. For each plasma model,
this paper puts forward the physics context, introduces the fundamental equations,
presents advantages and limitations, also from a numerical perspective, and gives some
illustrative examples. Only key information is given, to make clear the type of model
that best suits the purpose of each study, in addition to an extensive list of references
for complementary in-depth reading.
The organisation of this paper is the following. Sections 2 and 3 introduce kinetic
models, based on a statistical particle-in-cell / Monte-Carlo approach or the solution to
the Boltzmann equation, with special focus on the description of the electron kinetics.
Section 4 is dedicated to fluid models, based on the solution to the hydrodynamic
equations for the multi-fluid system of electrons, ions and neutrals that compose
plasma. Section 5 presents global (spatially-averaged) models, based on the solution
to the particle and energy rate-balance equations for the main plasma species, usually
including a comprehensive reaction chemistry. Section 6 introduce mesoscopic models for

Foundations of modelling of nonequilibrium low-temperature plasmas 3
plasma-surface interaction, which adopt either a deterministic approach or a stochastic
dynamical Monte-Carlo approach. Section 7 concludes with final remarks.
The text of the paper uses various acronyms and introduces many physical
quantities, which are not always defined when they appear for the first time in the
equations. Tables 1-3 present the list of symbols, the list of physical constants and the
list of acronyms, respectively, as used in the paper.
2. Plasma particle-in-cell simulation
A LTP is a mixture of charged and neutral particles. In a general way, each of these
species can be characterized by a particle distribution function, which depends on both
velocity space and real space coordinates. If there is also time dependence, then in
the most general case there are seven independent variables. In a nonequilibrium
LTP, there is no thermalization between the populations of the different species, each
having a particle distribution function with a different mean energy. The evolution
of the particle distribution functions is described by a Boltzmann transport equation
for each species [5] (see section 3), and these equations are coupled to each other
through collisional interactions. If this were not already a sufficiently complex problem,
charged species additionally interact with electromagnetic fields, which may be either
self-generated or externally applied. If these fields are to be consistent with the
particle distribution functions, then the set of Boltzmann equations must be coupled
to Maxwell’s equations. The solution of this system of equations is a challenge almost
always beyond the resources available to LTP physicists, so that simplifications must
usually be sought. Most methodologies for generating numerical solutions of systems
of partial differential equations replace the continuous functions of the mathematical
problem with a finite set of values, typically specified at a set of points defined by
dividing each coordinate axis into uniform intervals. If there are N
I
such intervals on
each axis, and the problem has d dimensions, then the number of values to be computed
is (N
I
)
d
. If N
I
= 100 (a modest number), then a single seven dimensional particle
distribution function involves the computation of some 10
14
values, which shows why
such calculations are not much attempted. Clearly, the most direct way to manage this
formidable number is to reduce the number of dimensions. Reducing the number of real
space dimensions simplifies the physical problem rapidly and drastically, but whether
or not this is acceptable depends on the objectives of the computation. Reducing the
number of velocity space dimensions is equally powerful, but has more subtle physical
consequences. An elegant approach that is discussed in detail in section 3 of this article
expresses the particle distribution function in velocity space (also called the velocity
distribution function) in terms of a spherical harmonic expansion. This method reduces
the number of velocity space coordinates from three to one (and, therefore, reduces
the computational burden by a factor ∼ 10
4
), often without much loss of physical
fidelity, at least where electrons are concerned. However, in principle, the spherical
harmonic approximation does not work well for ions, and may not be satisfactory for

Foundations of modelling of nonequilibrium low-temperature plasmas 4
electrons in regions of strong electromagnetic fields. In practice, the spherical harmonic
approximation is frequently combined with further approximations affecting the coupling
between charged particles and fields. For these reasons, numerical schemes that compute
the particle distribution functions without such simplifying assumptions are of interest.
In this context, we introduce particle-in-cell simulation.
Particle-in-cell (PIC) simulation adopts a different strategy for reducing the amount
of computation [6–8]. Instead of spanning each velocity axis with fixed points where the
values of the particle distribution functions are computed, a PIC simulation introduces
a set of particles that move through velocity and real space under the action of
electromagnetic fields. The physical particle distribution function is then expressed
as a sum over the set of computational particles. A high degree of accuracy can be
achieved with a surprisingly small number of particles. The real space axes are divided
into intervals as before, so that one can speak of real space as divided into cells, and
every particle is therefore located in such a cell, hence the name of the method. Many
calculations can be satisfactorily accomplished with around 100 particles per cell, so
the reduction in computational burden is again around 10
4
, which seems similar to a
spherical harmonic expansion. However, for reasons of stability and accuracy, the cell
sizes in PIC simulations are restricted by basic plasma properties such as the plasma
frequency and the Debye length. A simulation method based on a spherical harmonic
expansion usually adopts additional assumptions that avoid these constraints, so that
in practice a PIC simulation is appreciably more costly — probably by at least a factor
of ten in most cases, and sometimes much more.
2.1. Basic approach
This section will display the basic characteristics of the PIC method by discussing the
simplest case of a system with three dimensions — one real space, one velocity space
and time. The equations of motion for the charged particles can include relativistic or
quantum effects in a semi-classical approximation [7], but these are usually not needed
in LTPs, so the relevant formulation is:
dx
i
dt
= v
i
(1)
dv
i
dt
=
q
α
m
α
E(x
i
) , (2)
where here the index i refers to a particle and α a species (electrons or ions). Of course,
all particles of the same species have the same mass m
α
and charge q
α
. This formulation
(and the discussion below) refers to the simplest case where only one axis is considered
in both real and velocity space. More general formulations, including magnetic field
effects, are basically similar in structure, and can be found in the literature [6, 7]. In
the electrostatic case, the electric field E(x) = −dΦ(x)/dx (with Φ(x) the electrostatic
potential) is found by solving the Poisson equation
d
2
Φ
dx
2
= −
ρ

0
, (3)