Thermophysical properties of nitrogen plasmas under thermal
equilibrium and non-equilibrium conditions
Wei Zong Wang1,2, Ming Zhe Rong1, J. D. Yan2, A. B. Murphy3 , M. T. C. Fang 2, Joseph W Spencer2
1. State Key Laboratory of Electrical Insulation and Power Equipment, Xi’an Jiaotong University, Xi’an
Shaanxi 710049, People’s Republic of China
2. Department of Electrical Engineering and Electronics, The University of Liverpool, Brownlow Hill,
Liverpool L69 3GJ, UK
3. CSIRO Materials Science and Engineering, PO Box 218, Lindfield NSW 2070, Australia
Calculated thermophysical properties of nitrogen plasmas in and out of thermal equilibrium are presented. The
cut-off of the partition functions due to the lowering of the ionization potential has been taken into account,
together with the contributions from different core excited electronic states. The species composition and
thermodynamic properties are determined numerically using the Newton–Raphson iterative method, taking
into account the corrections due to Coulomb interactions. The transport properties including diffusion
coefficient, viscosity, thermal conductivity, and electrical conductivity are calculated using the most recent
collision interaction potentials by adopting Devoto’s electron and heavy particle decoupling approach,
expanded to the third-order approximation (second-order for viscosity) in the framework of Chapman–Enskog
method. Results are presented in the pressure range of 0.1 atm to 10 atm and in electron temperature range
from 300 to 40 000 K, with the ratio of electron temperature to heavy-particle temperature varied from 1 to 20.
Results are compared with those from previous works, and the influences of different definitions of the Debye
length are discussed.
1
I.
INTRODUCTION
Plasmas can be in either a local thermodynamic equilibrium (LTE) state or a non-LTE state depending on
the nature of the application and the discharge conditions. LTE requires that transitions between energy states
of the plasmas particles and chemical reactions are controlled by micro-reversible collisions with radiative
processes playing a negligible role1. In LTE plasmas, local gradients of plasma properties (temperature, density,
thermal conductivity etc) are small enough to allow particles to arrive at the equilibrium: the diffusion time is
significantly longer than the time that the particles take to reach equilibrium2. For optically thin, homogeneous
plasmas, the electron number density is an indicator of whether LTE is satisfied, according to the Griem
criterion3. Research into thermal plasmas under the LTE assumption has in recent years been mainly driven by
potential industrial applications and by the ever-increasing demand for existing plasma technology4-10. However,
it has become increasingly clear that the existence of LTE in plasmas is the exception rather than the rule. For
example, for atmospheric pressure plasmas, substantial deviations from LTE occur in the fringes of plasmas or
in the vicinity of walls or electrodes, which are regions with low electron densities2. Another well-known
example of non-LTE plasmas is microwave-sustained plasmas, where the inelastic collisions between electrons
and heavy particles are excitative or ionizing and do not increase the heavy-particle temperature to that of
electrons. The degree of deviation from LTE depends on the conditions11.
A two-temperature model can be used to describe non-LTE plasmas if chemical equilibrium is satisfied,
the two temperatures being the electron temperature (Te) and the heavy-particle temperature (Th). Since the
mass difference between the heavy particles is tiny compared to that between electrons and heavy particles, all
the heavy particles have the same temperature Th, which is sometimes called the plasma temperature or gas
temperature.
To understand the plasma characteristics and to model and optimize the performance of the plasma systems,
2
thermodynamic and transport properties of the plasma gas are required. The accuracy of the property database is
a key aspect in producing correct modeling results. Nitrogen is widely used in industrial equipment either as a
protective gas such as in sealed relays for aerospace applications12, where erosion of electrical contacts by
switching arc is a decisive factor for the lifetime and reliability of the device, or as a working medium such as in
plasma systems used for production of nanotubes13, nitride nanoparticles14, nitriding15, and space propulsion16.
There has been a significant amount of work on the calculation of the thermodynamic and transport properties of
nitrogen under LTE and non-LTE conditions. For example, Capitelli et al.17 and Murphy et al.18 published
theoretical calculations of transport coefficients for LTE nitrogen plasmas based on collision integrals that were
the best available at the time, but some of which have since been updated. Aubreton et al.
19
calculated the
transport properties of nitrogen in non-LTE for heavy-particle temperatures up to 15,000 K at atmospheric
pressure with a maximum non-equilibrium degree θ = Te/Th of 2, which is not a sufficiently wide range to cover
all thermal plasma applications. Colombo et al.20 presented their calculation of thermophysical properties of
pure nitrogen with a two-temperature model. The lowering of the ionization potential due to interactions
between plasma particles, whose influence on the chemical equilibrium concentration increases with the
non-equilibrium degree, was not considered in the calculation of the partition functions2. Some spectral states
were also not included in their calculation due to a lack of fundamental data. In addition, the contributions to the
total thermal conductivity from the chemical reactions should be taken into account through the diffusion
velocities, which are functions of ordinary and thermal diffusion coefficients, and this is not reported in
Colombo et al.’s paper.
In the present work, a two-temperature model with new features has been developed to calculate the
properties of non-LTE plasmas in which heavy particles and electrons still follow a Boltzmann energy
distribution. but with different temperatures. The calculation is performed for a wide pressure range of 0.1 atm
3
to 10 atm and a temperature range of 300 K to 40,000 K. Compared with previous work, the temperature and
pressure ranges are significantly extended. The shielding effect of the charged species is systematically
studied. The typical features of thermodynamic properties and transport coefficients under non-LTE are
carefully studied.
Section 2 describes the basic principles, including a novel numerical technique to determine the partition
function and species composition for two-temperature plasmas. The calculation of thermodynamic properties is
presented in section 3. The fundamental theory used to calculate transport coefficients and the intermolecular
potentials used to evaluate collision integrals are given in section 4. In section 5, the results computed at
atmospheric pressure are compared with available results in the literature, and a number of important features of
the present calculation as well as the influence of different pressures on properties are described. Finally,
appropriate conclusions are drawn in Section 7.
II.
PARTITION FUNCTIONS AND PLASMA COMPOSITION
A prerequisite to obtaining thermodynamic properties of LTE and non-LTE nitrogen plasmas is the
determination of the species composition; this is also the starting point to obtain the transport coefficients21. The
paper considers the following gaseous species: molecules N2, molecular ions N2+, atoms N, singly-, doubly- and
triply-ionized ions N+, N++ and N+++ and electrons e, and therefore five independent chemical reactions are
included in the calculation:
N 2  N+N , N  N + +e , N 2  N +2 +e , N +  N ++ +e , N ++  N +++ +e
The fundamental laws that determine the species composition in two-temperature plasmas are the mass
conservation of the elements, Dalton’s law, the law of mass action (the Saha equation and the Gulberg–Waage
equation) and electrical quasi-neutrality2. The relevant nonlinear equations are written in the following form:
Species conservation:
4
ν
'
 c si n si = C
(1)
i
'
Equation (1) expresses the mass conservation of the elements; n si
is the amount of species i containing
the element s in the plasma, csi is the stoichiometric coefficient of the element s appearing in the species i, and
C is the total amount of elements.
Dalton’s Law:
w
P + Δ P =  n i k Th + n e k Te
ie
ΔP=
1
w
24 0
(2)
2
 Z i ni
3
i 1
d
Equation (2) represents Dalton’s Law taking into account the Coulomb field modification; P and ΔP are
respectively the gas pressure and pressure correction resulted from the charged particles interactions22. The
latter, which should be taken account of for non-LTE plasmas, was neglected in previous studies1. λ d is the
Debye length defined later and w the number of species in the system. k is Boltzmann constant,  0 the
vacuum permittivity, Zi and n i respectively the charge number and number density of species i, and Te and
Th the electron and heavy-particle temperatures respectively.
Electrical quasi-neutrality:
'
(3)
'
 Zt n t -n e =0
t
Equation (3) describes electrical quasi-neutrality in the plasma; Z t and n 't are respectively the charge
number and number density of the charged species t.
The law of mass action:
E -ΔE I,r+1
 2m e πkTe 3/2
n e n r+1  Qint
=2  r+1
(
) exp(- I,r+1
)
int 
2
nr
kTex
Q
h
 r 
(4)
E
n A n B Qint
Qint 2πm A m B kTh 3 2
= A int B (
) exp(- d )
2
n AB
kTex
Q AB
mA B h
(5)
Equations (4) and (5) are respectively the Saha and Gulberg–Waage’s equations, which describe ionization
5
( A z+  A (z+1)+ +e ) and dissociation reactions ( AB  A+B ) in the form proposed by the van de Sanden et
r
al.22-24. Qint
i and h are the internal partition function of species i and the Planck constant. The subscript
indicates r-times ionized species or molecules. AB , A , and B respectively denote the reactant and the two
products of the dissociation chemical reactions, and Tex is the excitation temperature of the relevant chemical
reaction. The energies of formation and reaction excitation temperatures relevant to nitrogen plasmas are given
in table I.
It is assumed that translational, rotational and vibrational motions of the heavy species are governed by Th ,
but electronic excitation and translational motion of the electrons are described by Te . Hence, the partition
functions are as follows22:

Monatomic species
el
Qint
i  Qi (Te )
(6)
where Qiel (Te ) is the electron internal partition function of species i .

Diatomic and polyatomic species
el
vib
rot
Qint
i  Qi (Te ) * Qi (Th ) * Qi (Th )
(7)
where Qivib (Th ) , Qirot (Th ) are, respectively, the vibrational and rotational partition functions of species i .
TABLE I. Chemical reaction energy changes and the reaction excited temperature.
Chemical reaction Energy (kJ/mol) Reaction excited temperature Tex
N 2  N+N
N 2  N +2 +e
N  N + +e
N +  N ++ +e
N ++  N +++ +e
N +++  N ++++ +e
941.6
1509.5
1402.3
2856.0
4578.1
7475.0
Th
Th
Te
Te
Te
Te
The lowering of the ionization energy is taken into account using the following expression:
6
ΔE I,r+1 =(r+1)
e2 1
4πε 0 λ d
(8)
where λ d is the Debye length representing the shielding effect of charged particles. It can be seen from Equation
(8) that the ionization potential lowering is inversely proportional to the Debye length. Different methods are
used to calculate λ d in the literature. Aubreton et al.19 and Colombo et al.20 both only consider the shielding
effects of electrons, which have much smaller mass and higher velocities than heavy particles. The Debye length
taking into account the shielding effects of both ions and electrons has the form:
λ -2d =
w
z 2t n t 
e2  n e
 + 

ε 0 k  Te t=1,t  e Th 
(9)
The second term in the brackets is neglected when only the shielding effects of electons is considered.
Equation (9) implies that ions play an equally or even more important role in lowering the ionization
potential in a two-temperature plasma. A comparison will be given in section 5 to show the non-negligible
influence of Debye shielding in calculating the thermodynamic and transport properties.
Unless specifically noted, only the shielding effect of the electrons is included in the Debye length λ d , in
order to allow comparison with the work of Aubreton et al. 19and Colombo et al. 20.
A. Evaluation of Partition Functions
The partition function of a species, which establishes the link between the coordinates of microscopic
systems and macroscopic thermodynamic properties, is the product of its translational and internal partition
functions. The translation of particles is of a continuous nature while the internal states are of discrete nature.
The translational partition function of species i can be obtained by integration over all spatial and momentum
coordinates to give
Qitr =(
2πmi kTi 3/2
) V
h2
(10)
where V is the volume of the system and mi the mass of species i.
7
The internal partition functions for monatomic species consist of the summation over all possible energy
levels  n up to effective ionization energy E ieff .
ε n <Eieff
-ε n /kTe
e
Qint
i  Q i (Te )=  g n e
(11)
n
Fundamental atomic data on energy levels and degeneracy (statistical weight) are obtained from Moore25
and the most recent data of NIST26. The contribution from energy levels whose data are partially known or
completely unknown has also been taken into account using a method similar to that described by Milone and
Merlo27. The contribution of partially-known states was accounted for using information about their
neighboring levels, and that of the completely unknown levels, usually with high principal quantum numbers,
was estimated using the hydrogenic approximation based on the last principal quantum number. The calculation
was performed with an upper limit on the quantum number of n = 1000. The calculation stops when the lowered
ionization energy is reached. The lowering of ionization energy depends strongly on temperature, which
influences the choice of the cut-off value of the quantum numbers used in the calculation of the partition
functions. In order to minimize the discontinuities in the numerical derivatives of the partition functions, the
overlapping of the sub-groups of energy levels characterized by consecutive principal quantum numbers is
taken into account, following Ref. 28. A similar approach was adopted by Colombo et al.20. However, they did
not take into account the contribution of the different multiple excited states (core excited electronic states),
which correspond to the different electronic configurations of the corresponding ions. The energy states of the
nitrogen atom, the associated degeneracy, and the ionization limits used in the present work are taken from Ref.
27 and given in table II.
TABLE II. States of the atoms considered in our calculation of the internal partition function of nitrogen atom
atom
Elec.
Config.
Ground
State
Excited States
1s22s22p2(3P)nx(3)
8
Ionization potential χion
[cm-1]
1173450.0
1s22s22p3
4 o
1s22s2p4
4
S
2
o
3/2( DP )
N
P5/2(2DPS)
*
1s22s22p2(1D)nx(3)
1s22s22p2(1S)nx(3)
1s22s2p3(5So)nx(3)
1s22s2p3(3Do)nx(3)
1s22s2p3(2Po)nx(3)
132660.7
150032.1
164512.7
209582.9
226563.2
For a diatomic or polyatomic molecule, the internal partition functions can be expressed as follows
Qint
i =
1
c
vmax
J max
v
J
 g e exp(-ε*el /kTe )  exp(-ε vib (v)/kTh )  (2 J  1)exp(-ε rot (v)/kTh )
e
(12)
with the vibration energy of the v-th vibrational state and J-th electronic state computed as
 vib (v)
hc
1
1
1
1
 e (v  )  e xe (v  ) 2  e ye (v  )3  e ze (v  ) 4
2
2
2
2
 rot ( J )
hc
 Bv J ( J  1)  Dv ( J ( J  1)) 2
where  vib (v) is the vibrational energy of the vth vibrational state and  rot ( J ) the rotational energy of the
Jth rotational state, c a symmetry factor, v and J the vibrational and rotational quantum number respectively,
and other parameters are spectroscopically-determined constants obtained from Refs. 29 - 30. In equation
(11), the quantity  el instead of  el is used to represent the energy levels proposed by Bacri et al28. This takes
into account that the potential energy of the fundamental level (v = 0, J = 0) of the ground configurations should
be set to zero and this leads to a negative equivalent electronic energy term

 el
0  (1 / 2)e  (1 / 4)e xe 
(13)

Note that the difference between  el and  el
can lead to significant modifications to the values of the
internal partition functions of diatomic molecules at low temperatures, as can be seen in figure 1 for the nitrogen
molecular and molecular ion in the LTE state, in which our results are compared with those of Drellishak et al. 31,
who used  el . The deviation at high temperature is due to different numbers of molecular spectral states used in
the calculation.
9
FIG.1. Temperature dependence of internal partition functions of nitrogen molecular and its ion
B. Determination of Plasma Composition
Electrostatic interaction between charged particles affects the equation of state [see Eq. (2)], and the
lowering of the ionization potential in turn influences the generalized Saha equation. This is because the cut-off
of spectral states in the calculation of the partition functions depends on the lowering of the species ionization
potential, which is a function of the Debye length [see Eq. (9)]. This mechanism has been taken into account to
ensure rigorous calculations in an iterative procedure developed in the present work.
10
FIG.2. Numerical computation flow chart
.
The iterative solution for the particle densities at a given temperature, temperature ratio and pressure for
both LTE and non-LTE plasmas is represented by the flow chart in figure 2. The process starts with the
calculation of the first set of plasma composition based on the ground state only. This provides an initial
guessed solution for the iterative computation. The partition function values, the lowered values of the
ionization potential, and the nonlinear equation expression corrections are updated and Newton–Raphson
method is adopted for iterations until convergence of the plasma composition is obtained.
III. DETERMINATION OF THERMODYNAMIC PROPERTIES
After obtaining the partition functions of each nitrogen species, the calculation of the thermodynamic
11
properties proceeds in a straightforward manner, employing the following standard thermodynamic
relationships.
Mass density:
v
ρ=  mi n i
i=1
(14)
where mi is the mass of species i.
Internal energy:
e=
int
3k
1
k
2 lnQi
 n i Ti +  n i Ei +  n i Ti
2ρ i
ρ i
ρ i
Ti
(15)
where Ei is the formation energy of species i.
Enthalpy:
h=
int
5k
1
k
2  l nQi
 n i Ti +  n i Ei +  n i Ti
Ti
2ρ i
ρ i
ρ i
(16)）
Specific heat at constant pressure:
Cp 
h
Te
(17)
IV. TRANSPORT COEFFICIENTS AND COLLISION INTEGRALS
A. Determination of Transport Coefficients
Transport properties, namely diffusion coefficients, viscosity, thermal conductivities and electrical
conductivity, are calculated approximately using the classical Chapman–Enskog method, which assumes that
the particle distribution function is a first-order perturbation to the Maxwellian distribution 32-34.
Up to now there are two approaches to deal with the transport properties in plasma. Devoto35 and
Bonnefoi36 developed a simplified theory neglecting the collisional coupling between heavy species and
electron. Rat et al.37-39 proposed a theory to take into account the coupling between heavy species and electron
12
that resulted in a set of new expressions for transport coefficients and coupling terms in the mass, momentum
and energy flux definitions.
It should, however, be noted that a recent study of non-LTE transport properties based on a comparison of
the approaches of Devoto and Rat et al. shows that coupling between electrons and heavy species does not lead
to significant changes in the predicted non-equilibrium plasma transport properties, except for certain ordinary
diffusion coefficients40. No significant discrepancies occur in the total thermal conductivity (including the
reactive contribution), viscosity or electrical conductivity, even when they depend on the ordinary diffusion
coefficients. Besides, at high temperatures when ionization processes dominate in the mixture, the third-order
approximation for thermal conductivity and second-order approximation for viscosity are required to provide
calculation accuracy of 1%. Consequently, we have used the simplified approach of Devoto with a third-order
approximation for transport properties, except for viscosity for which the second-order approximation has been
adopted.
Thermal conductivity depends not only on the translation of the particles. but also is affected by processes
related to the internal energy changes and chemical reactions. Chemical reactions (namely dissociation,
ionization and recombination) lead to an additional heat flux. This introduces an additional reactive thermal
conductivity component re . An expression for the reactive thermal conductivity of high temperature gases and
their mixtures42, which follows the Van’t Hoff equations modified to reflect the nature of two-temperature
plasmas41 for monatomic gases, is used in this work.
The presence of internal degrees of freedom can affect the heat flux vector, and gives rise to an internal
thermal conductivity in , which has been derived using the Hirschfelder–Eucken approximation as follows 43:
w
x j Dii 1
)
j  e xi Dij
w
int   PDii / RTh (C pi  2.5 R) ( 
i e
(18)
13
where Cpi is the specific heat at constant pressure of the species i, Dii and Dij respectively the self-diffusion
coefficient and the binary diffusion coefficient between species i and species j. R, P，xi, and w respectively
denote the molar gas constant, pressure , mole fraction of species i, and the total number of species.
B. Evaluation of collision integrals
The complicated combinations of bracket integrals, which appear in the expressions for the transport
coefficients, can be represented as linear combinations of the collision integrals, which are defined as follows
32-34
:
ij(l , s ) 
kTij 
2
2 s 3 l
Qij ( g )d  ij
 exp( ij ) ij
2ij 0
(19)）
The transport cross sections Qijl ( g ) [see Eq. (19)] are defined by

Qijl ( g )  2  (1  cosl  )bdb
0
(20)
where b is the impact parameter. ij and Tij are the reduced mass and the reduced temperature of the
colliding molecules i and j , respectively given by
1
ij

Tij 
1
1

mi m j
(21)）
miT j  m jTi
(22)）
mi  m j
Here  ij is the reduced initial speed of the colliding molecules i and j , given by
 ij 
 ij
2 kT
g ij
(23)
where gij is the initial relative speed of the colliding molecules i and j .  , the angle by which the molecules
are deflected in the centre of gravity coordinate system, is given by
14

 


 2b 



dr / r 2

1  ij (r ) 12 ij gij2   b 2 r 2


rm
(24)

）
where the term rm is the outermost root of the equation (25) and ij (r ) is the potential energy of interaction
between the colliding particles.
1
ij ( r )
1
ij gij2
2

b2
rm2
0
(25)
In this paper, reliable collision integrals were obtained as described in the following sub-sections.
1.
Neutral–neutral interactions
For the N2–N2 and N2–N interactions, the tabulated collision integrals determined by recent ab-initio
calculations from quantum-mechanically derived potential energy surfaces were used in view of the fact the
data were consolidated by extensive benchmark comparisons to previous theoretical and experimental results
44-45
.
In the case of the N–N interactions, collision integrals were obtained by averaging the contributions of the
bound and repulsive states of the N2 molecular system. The Morse potential and exponential repulsive potential
were used to fit the relevant interaction potentials. The parameters used for the potentials were those compiled
by Capitelli et al. 46
2.
Ion–neutral interactions
For interactions between neutral species and ions, two kinds of processes should be taken into account,
purely elastic collisions and the resonant charge-exchange process which is of inelastic nature. For l odd (l=1
and 3) [see Eq. (20)], the latter carries a heavy weighting in determining the collision integrals. Considering the
15
elastic and inelastic processes, we follow the previous work of Murphy47 and estimate the total collision
integrals by the empirical mixing rule:
(l , s )
(l,s)
(l ,s )  (in )2  (el ) 2
(26)）
where the subscripts in and el denote the collision integrals derived from the inelastic and the elastic interactions,
respectively.
In the case of the elastic interaction, collision integrals for N+–N have been calculated using Morse and
repulsive potentials averaged with statistical weights considering interactions through 12 potential curves. The
relevant potential parameters were given in Ref.46, and the values of the collision integrals ij(1,1) , ij(1,2) , ij(1,3)
and ij(2,2) that were tabulated there were extended in the presented work to include ij(2,3) , ij(2,4) and
ij(2,5) , as required for the third-order approximation.
For the N+–N2, N2+–N2 and N2+–N
interactions, the phenomenological potential (an improvement of the
Lennard–Jones potential) developed by Capitelli and co-workers 48-50has been used. The potential parameters
were defined in and taken from Ref.49 and are given in table III.
TABLE III Parameters for ion-neutral interaction potentials49
Interaction

N+  N2
7.3650
N 2  N 2
 0  eV 
re (Å)
m
0.12679
3.071
4
8.0746
0.09657
3.454
4
N 2  N
6.9373
0.06871
3.358
4
N 2  N
6.9373
0.06871
3.358
4
For other ion–neutral interactions involving multiply-charged species, collision integrals have been
derived using the polarization potential51:
16
in (r )  (
1
4 0
)2
( Z i e) 2  n
(27)
2r 4
with Zi being the ion charge and  n the polarizability of neutral species. The values of polarizability of N2 and
N are 1.76 Å3 and 1.13 Å3 respectively.
It should be noted that the use of this potential can lead to an underestimation of the collision integrals;
however, since the neutral species will have very low concentration when the concentration of multiply-charged
species is significant, and vice versa, the interactions have only a minor influence on the transport coefficients in
the situations under consideration.
For interactions between a parent atom or molecule X and its ion X+ or X–, particularly for high energies,
the inelastic transport cross section can be evaluated from the charge transfer cross-section using 47
Ql ( g )  2Qex
(28)
and the charge transfer cross section can be approximated by the following representation:
Qex  ( A  B ln E ) 2
(29)
where E is the collision energy. The constants A and B can be obtained from experiment data and theoretical
calculations.
For N2+–N2 and N+–N interaction, the relevant constants A , B are taken from Ref. 46. It is assumed that
the charge-exchange cross section for collisions between unlike species (e.g. Y   X  X   Y ) is small
compared to the elastic collision integrals, so it is neglected. Collision integrals with even l are wholly
determined by the elastic interactions.
3.
Electron–neutral interactions
The collision integrals for interactions between electrons and neutral species have been calculated by
straightforward integration over energy assuming a Boltzmann distribution based on the transport cross sections
17
Q1 (E), Q2 (E) and Q3 (E) as functions of electron energy. When Q2 (E) and Q3 (E) are not available, different
techniques are used to estimate them.
For the e–N2 interaction, we followed the recommendation of Wright et al. 52. The momentum transfer
cross sections at lower energies recommended by Itikawa53 were adopted, and complemented by the differential
elastic cross-section (DCS) data between 0.55 and 10 eV from Sun et al. 54and those above 10 eV from Nickel et
al.
55
. The differential cross sections can be numerically integrated over all scattering angles to obtain the
transport cross sections together with the ratios of Q2 (E) and Q3 (E) to the momentum transfer cross section Q1
(E) as a function of the interaction energy:
 d
Ql  E   2 
0 d
sin θ(1  cosθl ) dθ
(30)
For the e–N interaction, the momentum transfer cross sections of Capitelli and Devoto 17 were corrected
based on the work of Thompson56. The ratios of Q2 (E) and Q3 (E) to the momentum transfer cross section Q1
(E) were obtained by using phase shift corrections as with that of Capitelli and Devoto 17.
4.
Charged species interactions
These interactions are described by a Coulomb potential screened at the Debye length by the presence of
charged particles; the effective collision integrals were calculated from the works of Mason et al. 57 and Devoto58.
There are different treatments of the role of ions in shielding. To assess the influence of ions, results with and
without ion shielding are compared in the present work for a clear indication for any future work. The respective
definitions of the Debye length are given in equation (9).
V.
RESULTS AND COMPARISONS
We present calculated compositions, thermodynamic properties and transport coefficients of nitrogen
plasmas under thermal LTE and non-LTE conditions (   Te Th = 1, 2, 3, 5, 10, 15, 20) in a wide pressure range
18
of 0.1-10 atm for the electron temperature range of 300–40,000 K, and compare our results with those available
in the literature.
A. Partition function and equilibrium composition
FIG. 3. Internal partition function of nitrogen atom under different degrees if non-LTE (solid line with symbols:
using Debye length from only electrons; dashed line with symbols: Debye length from electrons and ions)
FIG. 4. As Fig. 3, but for the monatomic nitrogen ion N+.
It has been shown in the present work that the inclusion of ions in the calculation of Debye length leads to
significant difference in the partition functions of atomic and ionic species, as shown in figures 3 and 4. For
19
example, at an electron temperature of 20,000 K with θ=5, the inclusion of both ions and electron in the
shielding Debye length contributes to a 0.1382 eV reduction in the ionization potential of nitrogen atom in
comparison with a value of 0.0529 eV due to electron shielding, which are respectively 0.95% and 0.36% of
the non-disturbed ground state ionization potential 14.53 eV. Its influence on composition is discussed in the
next section. The decrease in the partition function corresponds to a reduction of the number of excited energy
states. The internal partition function decreases from 16.091 to 9.255 correspondingly. The internal partition
functions of monatomic species depends strongly on the concentration of charged particles, which can lead to
lowering of the ionization potential and affect the cut-off behaviour of the internal partition functions of these
species. Including the screening effect of the ions in calculating the Debye length leads to a larger reduction of
the species ionization potential, so a smaller number of energy states are considered. Therefore, as shown for the
nitrogen atom and atomic ion in figures 3 and 4 respectively, the internal partition function for a given value of
θ is smaller than that obtained considering only the electrons.
Figure 5 shows the number density of plasmas species as a function of electron temperature under
different non-LTE degrees. Taking θ=1 as a reference case, the inclusion of ion shielding in the definition of
the Debye length alters its value by 29.3% compared to the definition that includes only the electrons. The
effect of this change on the composition is attenuated through the lowering of the ionization potential in the
partition function expression. As a consequence, the inclusion of ions in Debye length calculation does not
produce a significant effect on the composition in the LTE case. The situation however starts to change when
θ increases. At a given temperature, a large value of θ means a lower Th in comparison with the reference case,
thus a reduced shielding effect of ions and a larger reduction in ionization potential. This leads to a reduction
in the number of neutral particles and an increase the ion density (and also electron density because of
quasi-neutrality). On the other hand, a lower value of Th at a fixed pressure and Te means a higher neutral
20
particle density [see Eq. (2)]). The resultant densities of electrons, ions and neutral particles will depend on
the balance between these two mechanisms.
（a）
（b）
（c）
21
（d）
（e）
FIG. 5. Temperature dependence of the number density of different species in nitrogen plasmas under different
degrees of non-equilibrium at atmospheric pressure (Solid line & symbols: Debye length including only
electrons; dashed line & symbols: Debye length including electrons and ions) (a) N2, (b) N, (c) electron, (d) N+,
(e) N++.
The number density of various species as a function of temperature for different values of θ in Figure 5
shows several distinctive features, the most important of which can be described as follows:
(1) The dissociation and ionization processes are governed by the temperature of heavy particles and
electrons respectively, except for nitrogen molecular ionization and dissociation, for which the excitation
22
temperature is identical with Th. For θ = 1 and 2, the dissociation of molecules and the first ionization of atoms,
take place at around the temperatures Th = 6800 K and Te = 15 000 K, respectively. As the value of θ increases,
the dissociation occurs at a lower value of Th of 3000 K at θ = 10, probably due to an increase in energetic
electrons promoting the dissociation process, and the ionization is shifted to higher electron temperatures.
(2) The concentration of nitrogen molecules remains high at higher electron temperatures for high values of θ
[see Fig. 5a] because the dissociation excitation temperature (equal to Th) is still low.
(3) The peak values of nitrogen atom number density decreases [see Fig.5b] rapidly with increasing θ. For
values of θ>2, the number density of nitrogen atom is much lower than that of nitrogen ions [see Fig 5d] at the
same temperature. This is because atoms will be immediately ionized into ions by impact of high energy
electrons once the dissociation reactions occur. The maximum number density of charged particles in a plasma
system can be higher in non-LTE cases than in LTE. At atmospheric pressure, the largest possible charged
particle number density for the conditions considered in this work is obtained with θ =3 at Te=17 000 K [see
Fig.5c].
(4) The number density of the nitrogen molecular ion is not presented in Figure 5 because of its decreasing
mole fraction at fixed temperature as θ increases. For θ>2, the highest mole fraction of molecular ions at
atmospheric pressure is lower than 10-7, so their influence on the chemical equilibrium composition as well as
other properties can be neglected. This phenomenon can be explained by the decreasing excitation
temperature Tex =Th at a given electron temperature with the increase of θ.
(5) The influence of the different Debye length definitions on the number densities of species, particularly
charged species, becomes more pronounced as the temperature and θ increase. For example, for θ=10, an order
of magnitude difference for electrons exists at Te=33 000 K. This is a direct consequence of the reduced ion
shielding effect. The ionization degree of nitrogen molecules is lower when they dominate in the system. For
the second nitrogen ion, the influence of the different Debye length definitions occurs beyond the electron
temperature range considered in this work. Therefore, for both cases, their number densities are less affected
[see Fig 5a and 5e]. For a given θ, the threshold Te (in K) above which ion shielding starts to affect the
charged particle number density can be estimated by Te  11020  2171
.
B. Thermodynamic properties
Thermal non-equilibrium significantly affects the species composition, which in turn determines the
23
thermodynamic properties. The choice of definition of Debye length is only important at larger values of θ, and
only then for high electron temperatures for which ionization becomes significant.
FIG. 6. Temperature dependence of mass density of nitrogen plasmas under different degrees of non-equilibrium;
symbols as in Fig. 5.
Figure 6 presents variation of mass density as a function of electron temperature for values of θ of 1, 2, 3, 5,
10, 15, and 20. The mass density increases at a fixed electron temperature as the θ value rises as a result of the
delayed dissociation and ionization. At low heavy-particle temperatures, this corresponds to a higher number
density of N2, following the equation of state [see Eq. (2)]. The steep slopes for curves other than θ=1
correspond to surges in dissociation and ionization. It can be seen that the mass density is very sensitive to the
value of θ at low electron temperatures, for which no dissociation takes place and the change in density is
simply due to the change in Th , Th = Te /θ; the sensitivity decreases at higher Te when the ionization degree is
large and the influence of the electrons is dominant.
24
FIG. 7. Temperature dependence of enthalpy of nitrogen plasmas under different degrees of non-equilibrium
(Solid line & symbols: this work, Debye length including only electrons; dashed line & symbols: this work,
Debye length including electrons and ions; open symbols: Colombo et al.20 .
FIG. 8. Temperature dependence of internal energy of nitrogen plasmas under different degrees of
non-equilibrium; symbols as in Fig. 5.
25
FIG. 9. Temperature dependence of specific heat at constant pressure of nitrogen plasmas under different
degrees of non-equilibrium as Fig. 7.
The specific enthalpy, internal energy and specific heat are shown as a function of electron temperature for
different values of θ in Figures. 7–9. The specific enthalpy and the internal energy decrease for increasing θ
values at a given electron temperature.
For the plasma in LTE, a separate chemical reaction contributes towards the height of each peak in the
specific heat. The first peak corresponds to dissociation of nitrogen molecules, and the subsequent peaks to the
first and second ionization of nitrogen atoms. For non-LTE cases, it is difficult to separate the different
contributions; the dissociation and ionization contributions are superimposed due to the higher dissociation
temperature of nitrogen molecules.
The results for specific enthalpy and specific heat at constant pressure are in good agreement with data
reported by Colombo et al.
20
for θ=1,2,3. The slight discrepancies may be attributed to the different
computational accuracy of the internal partition functions, which lead to uncertainties in the plasma
composition and hence, the thermodynamic properties. As noted previously, the influence of multiple excited
states and the lowering of the ionization potential were not considered by Colombo et al.
The definition of the Debye length is found only to affect the thermodynamic properties significantly at
26
large values of θ and at high electron temperatures. This is consistent with its influence on the species
composition. A comparison for θ>3 is not possible due to a lack published data.
C. Transport properties
The thermodynamic properties and transport coefficients of thermal plasmas are prerequisite input data for
reliable numerical simulations of plasma behavior. Unfortunately, large discrepancies still remain in the values
of transport coefficients given by different authors. These partly result from the uncertainties in the gas
composition and approximations made in the calculations, but are mainly due to uncertainties in the values of
the intermolecular potentials from which the transport coefficients are evaluated. The most recent interaction
potentials are adopted in this work and therefore can provide more accurate transport coefficients, i.e. diffusion
coefficients, viscosity, thermal conductivity and electrical conductivity as shown in figures 10-17. We also
present comparisons with the results of Colombo et al.20. Colombo et al. used the same interaction potentials as
Murphy and Arundell18 for LTE nitrogen transport properties, and gave nearly identical values of the properties.
We also investigate the influence of the Debye length definition. As well as affecting the plasma
composition, we note that the inclusion of both electrons and ions in calculating the Debye radius, which
determines the screening distance, leads to a value of the Debye radius that is a factor of
2 / 2 smaller than if
only electrons are considered under LTE conditions. This gives a smaller value of the Coulomb collision
integrals, and therefore larger transport coefficients. The changes to the collision integrals have a much stronger
effect on the transport coefficients than the changes to the plasma composition.
1.
Diffusion coefficients
27
FIG. 10. Temperature dependence of electron thermal diffusion coefficients of nitrogen plasmas under different
degrees of non-equilibrium; symbols as in Fig. 5.
Because the number of ordinary diffusion coefficients describing a plasmas is large (one for each pair of
species), only the values of the thermal diffusion coefficient of electrons are presented here, although all
diffusion coefficients are calculated.
As shown in the simplified expression given in equation 31, the electron thermal diffusion coefficient
depends on the electron temperature, electron number density and the coefficients ai 0 ( ) , which are functions
of the collision integrals.
nm
DeT ( )  e e
2
2kTe
ae0 ( )
me
(31)
As discussed above, the different Debye length definitions have little influence on the electron number
density at the lower electron temperatures before the number density of electrons reach its maximum, as shown
in Figure 5c. At the same time, the interactions between charged particles are not dominant because the
ionization degree is low, so the change in the Coulomb collision integrals resulting from the different Debye
length definitions plays a negligible role. Therefore, the electron thermal diffusion coefficients are almost not
affected in this temperature range for any given value of θ, as shown in figure 10. For higher electron
temperature, for which ionization is significant and the charged interactions play an important role, the smaller
Coulomb collision integrals and higher electron densities when ion shielding is considered lead to higher values
of the electron thermal diffusion coefficients.
28
2.
Viscosity
FIG. 11. Temperature dependence of viscosity of nitrogen plasmas under different degrees of non-equilibrium;
symbols are as in Fig. 7.
Figure 11 compares our calculated temperature dependence of the viscosity of nitrogen in and out of LTE
with the results of Colombo et al. 19. The results in the case of Debye length including only electrons show
general agreement, except at the low heavy-particle temperatures for which interactions between neutral species
dominate. It is noted that the viscosity at a given temperature is, to a good approximation, proportional to a
weighted average of ij(2,2) for the pairs of species i , j that are present.
The collision integrals used in the present work for interactions between nitrogen atoms, from the work
of Capitelli et al.46, are in excellent agreement with those used by Colombo et al. (from Levin et al.59),
particularly in the heavy-particle temperature range in which nitrogen atoms are the predominant species. It is
apparent that the discrepancy in viscosity values up to the dissociation temperature of nitrogen (about 6800 K) is
)
)
)
a consequence mainly of differences in N(l , s-N
and N(l ,s-N
. The collision integrals N(l , s-N
and
2
2
2
2
(l , s )
obtained from exponential repulsion potentials used in Colombo et al.’s work are strongly
N
-N
2
2
underestimated compared to the recent ab initio calculations used here. At higher temperatures, for which the
N+-N interaction becomes important, the good agreement between our calculation and that of. Colombo et al. is
29
a consequence of the similar values of (l +, s ) , despite the different data source. At even higher temperatures,
N -N
for which interactions between charged species dominate, the deviation for non-LTE plasmas is attributable to
the different methods used to determine composition and Coulomb integral collision integrals. The deviations
are similar to those noted by Colombo et al. in comparing their results for argon plasmas with others in the
literature 20.
The viscosity first increases with temperature, and then decreases as the strong Coulomb interaction starts
to dominate. The maximum value of viscosity decreases as the non-equilibrium degree θ increases, and the
temperature at which this maximum occurs moves rapidly towards higher electron temperature as θ increases.
This is because ionization occurs at higher electron temperature, but lower heavy-particle temperature, as θ
increases [see Fig.5]
3.
Thermal conductivity
FIG. 12. Temperature dependence of total thermal conductivity of nitrogen plasmas in LTE. Straight line &
symbols: this work using collision integrals mentioned here; symbols: this work using collision integrals
described by Capitelli et al.46, and the work of Murphy and Arundell18.
Our values of the temperature dependence of total thermal conductivity of nitrogen in LTE are compared
with Murphy and Arundell’s results 18 in figure 12. The Debye length was calculated including only electrons.
30
)
)
As with the viscosity, the collision integrals N(l , s-N
and N(l ,s-N
used in the present work lead to a lower total
2
2
2
thermal conductivity at temperatures corresponding to the dissociation of nitrogen molecules. This deviation
indicates that not only the N–N interaction, but also the N–N2 and N2–N2 interactions, can play an important role
in determining the peak value of the total thermal conductivity. To confirm this, we calculated the thermal
conductivity using the collision integrals given by Capitelli et al.46, which are very similar to those used by
Murphy and Arundell18, and obtained values that agree well with those of Murphy and Arundell [see Fig. 12].
FIG. 13. Temperature dependence of translational thermal conductivity (λtr) of nitrogen plasmas under different
degrees of non-equilibrium; symbols are as in Fig. 7.
The calculated translational thermal conductivity (λtr) as a function of electron temperature for LTE and
non-LTE plasmas in the case of Debye length including only electrons is given in figure 13. Good agreement is
found with Colombo et al.’s data 20 obtained with the same Debye length definition, particularly in the middle
temperature range in which the N–N+ interaction dominates. This confirms the similarity of both the elastic and
charge transfer components of the N–N+ interatomic potentials used by Colombo et al. to our values, despite the
different data sources. The slight deviations at low temperature and high temperatures can be explained in
similar terms as the deviations in viscosity.
31
FIG. 14 Temperature dependence of reactive thermal conductivity of nitrogen thermal plasmas under different
degrees of non-equilibrium; symbols are as in Fig. 5.
Figure 14 presents the reactive thermal conductivity component as a function of electron temperature for
different values of θ. Since dissociation and ionization depend on the heavy-particle temperature and electron
temperature respectively, the reactive thermal conductivity shows a peak due to the dissociation of nitrogen
molecules (Th ≈ 6800 K) and the peaks due to the ionization of nitrogen atoms and ions (Te ≈ 15 000 K and
30 000 K). For θ = 1, the three peaks are clearly distinguished and the position of the dissociation peaks shifts
towards higher electron temperature as the value of θ increases.
The dependence of the reactive thermal conductivity (λre) on the ion shielding becomes stronger as the
non-equilibrium degree increases. This is mainly due to the difference in species densities caused by the altered
internal partition functions, as in the case of the thermodynamic properties.
32
FIG. 15. Temperature dependence of components of the thermal conductivity of nitrogen plasmas for θ = 3;
symbols are as in Fig. 5. λtot: total thermal conductivity; λtrh and λtre: translational components due to heavy
particles and electrons respectively; λre: reactive component; λin internal component.
TABLE III. Internal thermal conductivity and its percentage contribution to the total thermal conductivity,
for θ = 10.
Te (K)
4000
6000
8000
10 000
12 000
14 000
16 000
18 000
20 000
22 000
22 500
24 000
26 000
28 000
30 000
32 000
34 000
36 000
38 000
40 000
λint (W m-1 K-1) Contrib. (%)
0.0060
0.0116
0.0249
0.0730
0.1928
0.3961
0.6479
0.8827
1.0460
1.1190
1.1250
1.1140
1.0500
0.9330
0.7241
0.2250
0.0252
0.0079
0.0067
0.0075
33
19.79
26.39
38.68
61.46
78.92
87.40
91.21
92.76
92.40
87.08
84.52
70.24
44.87
21.72
6.65
0.49
0.08
0.09
0.08
0.08
The peak value of the internal component of the thermal conductivity (λin), arising from the transport of
internal energy, also increases and moves towards higher electron temperature as θ increases. For smaller values
of θ < 5, the internal component makes a negligible contribution to the total thermal conductivity as can be seen
in figure 15, in which the different components of the thermal conductivity for θ = 3 are shown. However, it is
noted that the internal thermal conductivity becomes significant for higher non-equilibrium degrees. The
variation of its absolute value and its contribution to the total thermal conductivity (λtot) as a function of electron
temperature is presented in table III for the case θ = 10.
FIG. 16. Temperature dependence of total thermal conductivity of nitrogen plasmas under different degrees of
non-equilibrium; symbols are as in Fig. 5.
The temperature dependence of total thermal conductivity is shown in figure 16. In the temperature range
in which dissociation and ionization take place, the reactive term makes a major contribution, while for hig
temperatures the translational contribution becomes important; internal thermal conductivity is always
negligible for low deviations from LTE but has to be taken into account at higher values of θ. The discrepancies
between the values obtained for the two Debye length definitions are attributable to differences in the reactive
component, as shown in figure 14.
4.
Electrical conductivity
34
FIG. 17. Temperature dependence of electrical conductivity of nitrogen plasmas under different degrees of
non-equilibrium; symbols are as in Fig. 7.
The electrical conductivity is proportional to the electron density; i.e., the ionization degree of the plasma.
This is evident from the first approximation expression for electrical conductivity:
 (1) 
3e2 ne

(1,1)
8 2 kTe me  n j e, j
j e
(32)
(1,1)
The collision integral e, j varies as a function of reduced temperature of electron and heavy particles,
which is approximately equal to the heavy-particle temperature. The temperature dependence of the electrical
conductivity is shown in figure 17. For each value of θ, the electrical conductivity first increases and then
slightly decreases, mainly due to the variation of the electron density.
Dissociation of nitrogen molecules, which is related to the heavy-particle temperature, does not occur until
the corresponding electron temperature is reached. This shifts the electrical conductivity curves towards higher
electron temperatures as θ increases.
The degree of non-equilibrium and the ions shielding influence the electrical conductivities through their
influence on the electron density. Further, the Debye length definition influences the Coulomb collision
integrals as discussed above, which makes a large contribution to the deviations shown in figure 17. 35
The electrical conductivity values presented here are in good agreement with those reported by Colombo et
al.20 when the same Debye length definition including only the contributions of electrons is used. The slight
discrepancy is attributed to the different electron–neutral interactions potentials and the method of
determination of the species composition.
D. The influence of pressure on properties
(a)
(b)
36
(c)
(d)
(e)
37
(f)
FIG. 18. Influence of pressure on electron mole fraction (a), specific heat at constant pressure (b), electron
thermal diffusion coefficients (c), viscosity (d), thermal conductivity (e) and electrical conductivity（f）of
nitrogen plasmas under different pressures 0.1atm, 1atm, 2atm, 3atm, 5atm, 10atm for θ=3; Symbols are as in
Fig. 5.
Figure 18 presents the influence of pressure on the chemical equilibrium composition, specific heat and
transport coefficients for pressures 0.1atm, 1atm, 2atm, 3atm, 5atm, 10atm for θ=3. Once again, results are
presented for the two Debye length definitions, whose influence on the properties has been discussed in
previous sections. According to Le Chatelier’s law, the increase of the pressure opposes changes to the
original state of equilibrium, so that dissociation and ionization at a given temperature are suppressed;
therefore, the relevant properties curves are shifted to a higher electron temperature as the pressure increases.
For viscosity [see Fig. 18 d], the temperature position of the maxima (Tmax) can be taken as the boundary
between a weakly-ionized gas and a plasma controlled by interaction between charged particles, and also the
point at which deviations in the properties begin to occur as a result of the differences in the charged particle
collision integrals caused by the different Debye length definitions.
For specific heat at constant pressure and thermal conductivity [see Fig. 18 b, e], the peaks that are related
to particular chemical reactions move towards higher electron temperature. The electron thermal diffusion
coefficients and electrical conductivity are directly dependent on the electron density, i.e. the ionization degree
of the plasma. Their values decrease as the pressure increases at electron temperatures below around 17 000 K,
38
as shown in figure 18 (c), (f). This is because the ionization temperature increases with pressure, so the
ionization degree is lower at a given temperature. Above around 17 000 K, the electron density, and thus the
electron thermal diffusion coefficients together with electrical conductivity, increase with pressure.
VI. CONCLUSIONS
In this paper, a considerable effort has been devoted to the calculation of partition functions, species
composition, thermodynamic properties and transport coefficients of nitrogen plasmas in and out of thermal
equilibrium over a wide temperature range (300 to 40 000 K) and pressure range (0.1-10atm), assuming
chemical equilibrium. These data are required for the computational modelling of many plasma applications.
The plasma composition was determined by iterative numerical solution of the nonlinear equation system
based on fundamental principles taking into account the lowering of ionization potential and contributions
corresponding to different core excited electronic states, and using partition functions derived from the most
recent spectroscopic data. The thermodynamic properties, obtained from the composition and internal partition
functions, have been presented in detail. For the calculation of transport coefficients, collision integrals obtained
from recent intermolecular interaction studies were used. Results obtained using two different shielding
distances (i.e. Debye length definitions), considering only electrons or all charged species, were compared.
Comparisons with data from the literature have also been carried out, and generally good agreement has been
found with the recent data.
It was found that with the increasing degree of non-LTE, it is possible to arrive at a situation where the
electron number density is much higher than that in LTE. An important finding is that the ionization potential
reduction due to the Coulomb interaction of charged species cannot be ignored for a non-LTE degree of θ = 3
or above for nitrogen plasmas. This was obtained by comparing the composition taking into account and
neglecting the influence of the ion shielding effect on the lowering of the ionization potential., The changes in
39
the chemical equilibrium composition can also lead to particular features in the thermophysical properties, such
as peaks in the viscosity and reactive thermal conductivity and the rapid increase in electrical conductivity,
being shifted to higher electron temperature. Properties became sensitive to the choice of Debye length
definition in the case of large θ. This is partly due to the differences in plasma composition as a consequence of
the altered internal partition functions, but predominantly due to the different collision integrals for interactions
between charged species, which are affected by the different screening distance used in evaluating the Coulomb
potential.
The results presented here are expected to be more accurate than those previously published. They will
serve as reliable reference data for computational simulation of the behaviour of nitrogen plasmas.
ACKNOWLEDGMENTS
This work was supported by the Chinese Government Scholarship program for postgraduates and Dual
Collaborative PhD Degree Program between Xi’an Jiaotong University and University of Liverpool.
40
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