Atmospheric direct current (dc) corona discharge from thin wires or sharp needles has been widely used as an ion source in many devices such as photocopiers, laser printers, and electronic air cleaners. Existing numerical models to predict the electron distribution in the corona plasma are based on charge continuity equations and the simplified Boltzmann equation. In this paper, negative dc corona discharges produced from a thin wire in dry air are modeled using a hybrid model of modified particle-in-cell plus Monte Carlo collision (PIC-MCC) and a continuum approach. The PIC-MCC model predicts densities of charge carriers and electron kinetic energy distributions in the plasma region, while the continuum model predicts the densities of charge carriers in the unipolar ion region. Results from the hybrid model are compared with those from prior continuum models. Superior to the prior continuum model, the hybrid model is able to predict the voltage–current curve of corona discharges. The PIC-MCC simulation results also suggest the validity of the local approximation used to solve the Boltzmann equation in the prior continuum model.

1. See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/216054259
A Hybrid Model to Predict Electron and Ion Distributions in Entire
Interelectrode Space of a Negative Corona Discharge
Article in IEEE Transactions on Plasma Science · February 2012
DOI: 10.1109/TPS.2011.2174806
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2. IEEE TRANSACTIONS ON PLASMA SCIENCE, VOL. 40, NO. 2, FEBRUARY 2012 421
A Hybrid Model to Predict Electron and Ion
Distributions in Entire Interelectrode Space
of a Negative Corona Discharge
Pengxiang Wang, Fa-gung Fan, Francisco Zirilli, and Junhong Chen, Member, IEEE
Abstract—Atmospheric direct current (dc) corona discharge
from thin wires or sharp needles has been widely used as an ion
source in many devices such as photocopiers, laser printers, and
electronic air cleaners. Existing numerical models to predict the
electron distribution in the corona plasma are based on charge
continuity equations and the simplified Boltzmann equation. In
this paper, negative dc corona discharges produced from a thin
wire in dry air are modeled using a hybrid model of modified
particle-in-cell plus Monte Carlo collision (PIC-MCC) and a
continuum approach. The PIC-MCC model predicts densities of
charge carriers and electron kinetic energy distributions in the
plasma region, while the continuum model predicts the densities of
charge carriers in the unipolar ion region. Results from the hybrid
model are compared with those from prior continuum models.
Superior to the prior continuum model, the hybrid model is able
to predict the voltage–current curve of corona discharges. The
PIC-MCC simulation results also suggest the validity of the local
approximation used to solve the Boltzmann equation in the prior
continuum model.
Index Terms—Corona plasma, electron, Monte Carlo methods,
particle-in-cell (PIC).
I. INTRODUCTION
CORONA DISCHARGES have attracted much attention
in recent years due to their broad applications [1]. The
corona discharge is a weakly luminous discharge that usually
takes place at or near atmospheric pressure. It is often produced
between two asymmetrical electrodes with an electric potential
difference. The discharge electrode usually has a small radius
of curvature, e.g., a sharp point or a thin wire, and the passive
electrode usually has a much larger radius of curvature, e.g.,
a flat plate or a cylinder. The polarity of the corona is either
positive or negative depending on the relative electric potential
applied to the discharge electrode with respect to the passive
electrode [2]. Fig. 1 shows a direct current (dc) negative corona
Manuscript received May 20, 2011; revised July 25, 2011 and September 20,
2011; accepted October 28, 2011. Date of publication December 8, 2011; date
of current version February 10, 2012. This work was supported in part by Xerox
Corporation through a UAC grant. The work of P. X. Wang was supported by
UWM Dissertation Fellowship.
P. Wang and J. Chen are with the Department of Mechanical Engineering,
University of Wisconsin–Milwaukee, Milwaukee, WI 53211 USA (e-mail:
pwang@uwm.edu; jhchen@uwm.edu).
F. Fan is with the Mechanical Engineering Sciences Laboratory, Fairport, NY
14450 USA (e-mail: fagung@gmail.com).
F. Zirilli is with the Xerox Research Center Webster, Xerox Corporation,
Webster, NY 14580 USA (e-mail: Francesco.Zirilli@xerox.com).
Color versions of one or more of the figures in this paper are available online
at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TPS.2011.2174806
Fig. 1. Sketch of a negative wire–cylinder corona discharge. The dashed line
indicates the outer boundary of the plasma region. Note that the ionization
boundary is thinner than the plasma boundary for negative coronas (not to
scale).
discharge in wire–cylinder geometry. Corona initiates when
the electric potential across the electrode gap is sufficiently
high that gas ionization occurs near the wire. An active corona
plasma region forms adjacent to the discharge wire. The highly
nonequilibrium corona plasma contains electrons at a few elec-
tronvolts and heavy species (atoms, molecules, and ions) near
room temperature. A short distance away from the discharge
electrode, the electric field is insufficient to sustain the ioniza-
tion, and unipolar ions of the same polarity as the discharge
wire drift into this region to fill the interelectrode space. In
contrast to the uniform positive corona discharge, negative
corona discharges appear as discrete points or tufts (Trichel
pulses) along the wire [3], [4]. At voltages near the corona onset
voltage, only a few tufts appear. They are irregularly spaced
along the wire and preferentially appear at imperfections on
the surface. As the voltage is increased, the number of tufts
increases, and the distribution of tufts becomes more uniform,
which is the regime considered by this work.
An important application of atmospheric dc corona discharge
is the production of unipolar ions for electrostatic charging
of various objects. For many decades, photocopiers, laser jet
printers, and electrostatic precipitators have relied on the dc
corona discharge to charge surfaces or particulates [5], [6]. In
recent years, corona plasmas have been widely used to generate
nanomaterials [7]. In addition to useful ions, energetic electrons
produced in the corona discharge have led to undesirable ozone
production [8]–[11] and deposition of silicon dioxide on the
0093-3813/$26.00 © 2011 IEEE

3. 422 IEEE TRANSACTIONS ON PLASMA SCIENCE, VOL. 40, NO. 2, FEBRUARY 2012
discharge electrode [12], [13]. The authors have previously
developed a comprehensive numerical model to predict the
distribution and the production rate of ozone in a dc corona
discharge along a thin wire in both dry air [14]–[16] and humid
air [17]. The model combines predicted electron distributions
with plasma chemistry and transport phenomena [18], [19]. In
addition to successfully predicting the ozone production rate in
the dc corona, the model explains the experimental data and
unveils the underlying mechanisms for ozone production in a
corona plasma environment.
However, the existing numerical model [18], [19] to predict
the electron distribution in the corona plasma is based on charge
continuity equations and the simplified Boltzmann equation, in
which a local approximation is used to independently solve the
electron number density distribution and the electron kinetic
energy distribution. This particular approach depends on the
validity of the local approximation [20], [21]. Also, this fluid
approach is not applicable when the size of the discharge
electrode or the discharge gap is comparable to or much smaller
than the mean free path (mfp) of electrons and ions in the
corona discharge, for instance, in the case of corona discharges
from nanostructures. A kinetic-theory-based method such as
Monte Carlo simulation and direct solution of the Boltzmann
equation should be used in this case. Both particle-in-cell (PIC)
[22], [23] and Monte Carlo collision (MCC) [24]–[26] methods
and the hybrid of PIC and MCC [24], [27]–[33] have been
widely used in modeling various plasmas except weakly ionized
corona plasmas. In this paper, we report on the modeling of
negative dc corona discharges from a thin wire in dry air using a
coupled method of PIC-MCC and continuum approach. Results
from the hybrid model are compared with those from prior
continuum model for electron and ion distributions. In partic-
ular, the hybrid model will be used to predict charge carrier
distributions in the whole interelectrode space, which provides
additional valuable information, such as the voltage–current
(V –I) curve, that the prior continuum model cannot provide.
II. HYBRID PIC-MCC AND CONTINUUM MODEL
A. Computational Domain
The negative corona discharge from a wire with a radius of
63.5 μm is studied, and the discharge gap (distance between
two electrodes) is 2 mm. The computational domain (Fig. 2) for
the simulation is the entire interelectrode space bounded by the
discharge electrode surface (r = a) and the grounded electrode
(r = ro), which includes both the corona plasma region and
the unipolar ion region. A modified PIC-MCC model is used
to solve the electron distribution in the corona plasma region,
while a continuum model is used for solving the ion distribu-
tions in the plasma region and all species in the unipolar ion
region. The ith particle at location ri is indicated by the blue
solid circle. The cell containing the particle is bounded by two
consecutive nodes at Rj and Rj+1. The distributions of charge
carriers in the discharge are assumed to be 1-D, which is the
same as that used in our prior continuum model to facilitate the
comparison.
The total number of cells was determined by the required
accuracy of the electric field. In the present study, the electric
Fig. 2. Computational domain for the hybrid model. The solid blue circle
indicates a particle present at ri in a cell bounded by Rj and Rj+1.
field distribution without space charges solved by the numerical
model was compared with that obtained from an analytic solu-
tion. When the difference between electric field distributions
obtained by the two methods was within acceptable tolerance,
then the cell size was employed in the modeling. The compu-
tational domain (2 mm) was divided into 900 000 cells along
the radial direction with a cell length of Δr. This cell size
is much smaller than the Debye length of the system, which
could lead to greater numerical noise. In order to reduce the
numerical noise, number densities of electrons in the modified
PIC-MCC model were the average values over each cell in a
period after the total number of electrons started to fluctuate in
a small range.
B. Algorithm
A computational flowchart of the hybrid scheme is shown
in Fig. 3. The initial field distribution is set by solving the
Laplace’s equation with an applied voltage and geometrical
parameters of the computational domain. The PIC-MCC is
then used to obtain distributions of electrons, positive ions,
and negative ions (ne, np, and nn, respectively) with the
initial field distribution and an assumed number density of
electrons at the wire surface (
ne_bd). With known ne, np,
and nn in the corona plasma region, boundary conditions
of the continuum model are determined, and the continuum
model can be solved. After the continuum model calculation
is performed, distributions of charge carriers in the whole
space between two discharge electrodes are obtained. There-
after, the field distribution with space charges can be updated
by solving the Maxwell’s (Poisson’s) equation. The effect of
space charges on the electric field is relatively small com-
pared with the effect of electric field on the space charges.
Therefore, when the number of electrons fluctuates within a
small range, the electric field distribution tends to be stable.
The computed stable field strength on the wire surface (Es)
is then compared with that estimated by Peek’s formula (Ep)
[19], [34]
Ep = 3.1 × 106
δ
1 +
0.0308
√
δr
(V/m) (1)

4. WANG et al.: HYBRID MODEL TO PREDICT ELECTRON AND ION DISTRIBUTIONS 423
Fig. 3. Computational flowchart for implementing the hybrid PIC-MCC and
continuum model.
Fig. 4. Flowchart for an explicit PIC scheme with MCC handler.
where
δ =
T0P
TP0
and r is the electrode radius in meters, T0 and P0 are the
standard temperature and pressure, respectively, and T and P
are the actual temperature and pressure of air, respectively. If
the difference between Es and Ep is not acceptable, then a new
value of
ne_bd is assumed to start a new computation loop. The
procedure is performed until the difference between Es and Ep
is less than 0.1%.
Implementation details of the PIC-MCC method have been
described by Birdsall and Langdon [22] and Vahedi and
Surendra [29] and, thus, are briefly summarized here. Fig. 4
shows a flowchart of the PIC-MCC computational scheme. The
basic idea of PIC-MCC is to use a small number of superpar-
ticles to represent a large number of real particles (electrons,
positive ions, and negative ions) in the corona plasma. The su-
perparticles are assumed to have the same charge-to-mass ratio
as real particles. Some seed electrons are released into the cells
in the computational domain from the boundary with a spatially
uniform initial number distribution and a normal distribution in
the velocity space. Like real particles, superparticles move in
the electric field following Newtonian equations of motion
dri
dt
= vi (2)
m
dvi
dt
= F(ri) (3)
where ri and vi are the position and the velocity of particles,
respectively, subscript i indicates the particle index, t is the
time, and m is the particle mass; in the present model, the mass
of electrons is 9.1094 × 10−31
kg, the mass of ions is the mean
value of nitrogen and oxygen molecules (4.8106 × 10−26
kg),
and F = qE is the electrostatic force applied on the particle
with E and q as the electric field and the charge carried by the
particle.
Because the mass of ions is much larger than that of elec-
trons, ions move much slower in the electric field. It is very time
consuming to track ionic species in the PIC-MCC model from
one boundary to the other. In the parametric studies, we found
that distributions of ions move forward always with a stable
front. With this characteristic, the value of the stable front can
be used as the boundary condition to solve distributions of ions
using the continuum model. Therefore, the modified PIC-MCC
model is used to track all electrons within the plasma region
and only selected ions at the flow front. With this approach, the
distribution of ions can be obtained during the electron transit
time instead of the ion transit time. Thus, the computation time
is significantly reduced.
The electrostatic force is derived from the electric potential
calculated through Maxwell’s equation
∇2
φ = −e(np − ne − nn)/ε0 (4)
E = −∇φ (5)
where φ is the electric potential, ε0 is the permittivity of
free space, and e is the elementary charge. The charge carrier
densities (ne, np, and nn) in each cell are derived from the
particle positions ri, by a weighting technique. In cylindrical
system, r is the position along the radial axis. For example, a
superparticle at ri is assumed as a charge cloud with charge nc,
which is assigned between the two neighboring cells RJ and
RJ+1 with subscripts J and J + 1 as the cell index (Fig. 2)
nJ = nc
RJ+1 − ri
Δr
(6)
nJ+1 = nc
ri − RJ
Δr
. (7)
The electric field at each node point (cell boundary) is then
obtained from Maxwell’s equation. Since the particle may fall
anywhere in the mesh, the electrostatic force acting on the

5. 424 IEEE TRANSACTIONS ON PLASMA SCIENCE, VOL. 40, NO. 2, FEBRUARY 2012
TABLE I
ELECTRON-IMPACT REACTIONS CONSIDERED IN THE PRESENT MODEL
particle is interpolated using the electric field at the nearest node
points by inverse weighting
E(ri) =
RJ+1 − ri
Δr
EJ +
ri − RJ
Δr
EJ+1. (8)
After the electrons move one time step (Δt) following
Newtonian equations of motion [(2) and (3)], they are ready for
collision with neutral molecules. The collisions are tracked with
the Monte Carlo scheme. The spatial information of charged
particles is recorded to calculate the new electric field distribu-
tion by solving Maxwell’s equation in a new computation loop.
In the current modified PIC method, the electric field is solved
at each period of time (multiple time steps) instead of each PIC
loop to accumulate the particles in each cell. The average value
of particle number is then used to compute the number density
to obtain the source term in Maxwell’s equation.
In the MCC scheme, oxygen and nitrogen molecules are
assumed to be the background species, and their number den-
sities are assumed to be constant. The major electron-impact
reactions considered in the model are shown in Table I. These
Fig. 5. Oxygen cross-sectional data used in the model.
Fig. 6. Nitrogen cross-sectional data used in the model.
electron-impact reactions and the associated collision cross
sections used are the same as those used in the prior model
[18]. The total number of reactions accounted for in this work
is 39, including elastic collisions for N2 and O2, 20 excitations
for N2 and 11 excitations for O2, dissociations for N2 and O2,
ionizations for N2 and O2, and two attachments for O2. A
complete compilation of collision cross-sectional data can be
found in [35] and [36]. Figs. 5 and 6 show the collision cross-
sectional data set used for O2 and N2 in this model. Of course,
there are many more electron-impact reactions occurring in the
atmospheric corona discharge, such as step ionizations, many-
body collisions, and attachment and detachment processes,
which, in general, are also important to describe plasma prop-
erties. However, due to the extremely small ionization degree
in the corona discharge, effects of these secondary reactions on
the corona plasma properties are negligible [18]. Since ionic
species do not have sufficient energy to contribute to ionization,
collisions between ions and neutral molecules do not affect the
distribution of electrons and are thus neglected.
Considering the stability and the accuracy of the PIC-MCC
simulation, the time step should be chosen carefully [22], [33],
[37]. On one hand, larger time steps lead to particles moving
out of the domain boundary without any collisions and thus
result in inaccuracies. On the other hand, smaller time steps
make the computation more expensive. In the present work, the
time step is chosen as Δt = 5 × 10−14
s after a series of trial
studies. This value agrees well with the selection in the study by
Hong et al. [37].

6. WANG et al.: HYBRID MODEL TO PREDICT ELECTRON AND ION DISTRIBUTIONS 425
For the continuum model, continuity equations of ne, np, and
nn are solved from the following equations, and more details
about the continuum model can be found in [18] and [19]:
d(rneμeE)
rdr
= (α − β)neμeE (9)
d(rnpμpE)
rdr
= −αneμeE (10)
d(rnnμnE)
rdr
= βneμeE. (11)
C. Boundary Conditions
The electrodes in the discharge system are equipotential,
which results in the Dirichlet boundary conditions for the elec-
tric potential. In the hybrid method, the boundary conditions for
Maxwell’s equation (3) are the applied voltage at the discharge
electrode
V (a) = VHV (12)
V (ro) = 0 at the grounded electrode. (13)
Only one boundary condition is required for each species
density at the corona discharge electrode (electron and negative
ion) or at the plasma boundary (positive ion). Since the number
density of positive ions in the unipolar ion region of a negative
discharge is negligible, the boundary condition for positive ions
is defined at the plasma boundary to save computational time.
Boundary conditions for positive and negative ions are their
densities in the background neutral air. In this paper, these two
densities are assumed as zero since densities of both positive
and negative ions in the atmosphere are much lower than those
in an active corona plasma region
np(rp) = 0 at the plasma boundary (14)
nn(a) = 0 at the discharge electrode. (15)
However, the electron density is strongly coupled with the
electric field in a corona discharge. The Katpzov hypothesis
suggests that the electric field at the discharge electrode in-
creases proportionally with the increasing voltage below the
corona onset but will preserve its value after the corona is
initiated [38]. The electric field on the surface of the corona
electrode is constant and equal to a value derived by Peek [39],
[40]. Peek’s formula (1) is used to determine the threshold
strength of electric field for the corona onset at the discharge
electrode. Lowke and D’Alessandro studied the onset corona
field and showed good agreement of Peek’s formula with exper-
iments in cylindrical geometries [41]. Therefore, the boundary
condition for electrons is replaced by the additional condition
for the field strength on the discharge electrode surface Es =
Ep. One boundary value of ne(a) generates corresponding
distributions of the space charges which affect the field strength
at the wire surface Es. The value of ne(a) that makes Es = Ep
is the boundary condition of electrons. This approach provides
an indirect boundary condition for electron number density.
Iterations will be performed until the total number of electrons
Fig. 7. Comparison of voltage–current curves fitted from modeling results
using (solid curve) the hybrid model and (dashed line) Cooperman’s formula.
fluctuates within a small range and the electric field at the
discharge electrode is sufficiently close to the value predicted
by Peek’s formula.
The modified PIC-MCC model and the continuum model are
coupled at the plasma boundary, which requires the thickness
of the plasma region. The thickness of the plasma region is
defined as the radius at which the reduced field E/N equals
80 Td for negative plasmas [19]. In the present work, the
thickness of the plasma region is determined each time after
the Maxwell’s equation is solved and the field distribution is
obtained. Since positive ions are present only within the plasma
region, it is not necessary to compute the number density
of positive ions outside the plasma region. For electrons and
negative ions, the boundary conditions for the continuum model
are the number densities computed from the PIC-MCC model.
D. Computational Hardware and Parameters
The simulation was performed on a supercomputer cluster
of IMB LS21 at the Minnesota Supercomputing Institute. Typ-
ically, in a 32-processor parallel computation, for a case with
total number of superelectrons fluctuating between 1.4 × 106
and 1.5 × 106
, it took about 48 h to complete 5 × 105
time
steps (Δt = 5 × 10−14
s) before a converged electric field
was obtained for a negative corona. The computing time is
consumed mainly by the PIC-MCC model.
III. RESULTS AND DISCUSSION
In order to obtain the relationship between voltage (V ) and
current density (J), different values of the applied voltage
are computed for a = 63.5 μm wire: −3.2, −3.5, −3.8, and
−4.2 kV. At each applied voltage, one value of the current
density is obtained by performing the simulation with hybrid
model following the procedure described in Section II. A
voltage–current curve can be fitted from modeling results by the
hybrid model (solid line in Fig. 7). It can be seen that the corona
current increases with the increasing voltage, which agrees rea-
sonably well with the result predicted by Cooperman’s formula
[42] (dashed line in Fig. 7)
Jtot =
4πεoμpV (V − Vi)
d2 ln
d
a
(16)

7. 426 IEEE TRANSACTIONS ON PLASMA SCIENCE, VOL. 40, NO. 2, FEBRUARY 2012
Fig. 8. Electric field distribution in the negative corona discharge with V =
−4.2 kV and the effect of space charge on the electric field distribution (result
obtained by solving the Laplace’s equation). The wire surface is at 0 µm, and
the grounded electrode is at 2000 µm.
Fig. 9. Number density distributions of charge carriers in the negative corona
discharge with V = −4.2 kV. The position of the plasma boundary is indicated
at rp = 382 µm. The wire surface is at 0 µm, and the grounded electrode is at
2000 µm.
where Jtot represents the total corona current density, Vi =
aEp ln(d/a) is the starting voltage for the corona discharge, μp
is the mobility of the positive ions, a is the wire radius, and d is
the space between two electrodes. Compared with Cooperman’s
formula, the curve fit from the current data shows a stronger
nonlinear characteristic and predicts a slightly higher current
for a given voltage. Cooperman’s formula considers the case
when the electrostatic field (zero space charge) is much larger
than the field generated by the space charge alone. This is true at
a low discharge current (or voltage). At a high discharge current
(or voltage), more space charges are generated, and the electric
field generated by the space charge could be significant and
Cooperman’s formula tends to underestimate the corona current
at a higher voltage as shown in Fig. 7.
Distributions of the electric field and charge carriers in
the whole interelectrode space with an applied voltage of
−4.2 kV are shown in Figs. 8 and 9, respectively. From Fig. 8,
it can be seen that, at a higher voltage, the space charges
distort the electric field distribution compared with the case
of without space charges. This makes it possible to keep
the electric field near the inner electrode close to the value
predicted by Peek’s formula (convergence criterion). At the
wire surface, the densities of electrons (ne = 8.78 × 109
m−3
)
and negative ions are at the boundary values. In the region
Fig. 10. Comparison of the number density distributions of charge carriers in
the negative corona plasma computed from the continuum model and the hybrid
model. Solid curves are the number densities obtained by the hybrid model, and
the dashed curves are those obtained by the prior continuum model.
close to the wire surface, the electron number density increases
with increasing distance from the wire surface, similar to the
tendency obtained from the prior continuum model (Fig. 10).
The increase in electron number density is due to the larger
rate of ionization than attachment. In Fig. 9, beyond the plasma
boundary (rp = 382 μm), the electric field is not strong enough
to support the ionization. In the unipolar ion region, most of the
effective collisions between electrons and neutral molecules are
the attachments that produce new negative ions by consuming
electrons. Therefore, the number density of negative ions nn
still increases with increasing radial position (r) in the unipolar
ion region although the increasing r may lead to decreasing
number density in the case of a constant particle number. In
Fig. 9, the decreasing ne is the result of not only the increasing
r but also the attachment collisions. Ionization occurs mainly in
the plasma region, so np increases from the boundary value at
the plasma boundary to the peak value at the wire surface.
Distributions of number densities of space charges in the
plasma region computed from the PIC-MCC part of the hybrid
model are compared with those from the prior continuum
model [18], [19] as shown in Fig. 10. Since the number of
particles computed from the PIC-MCC model varies with time,
the electron number densities shown in the figure are the
time-averaged values from the time when the total number of
particles is stable. The difference between the PIC-MCC and
the prior continuum models may be partly attributed to the
statistical error in the MCC process. In MCC, the electrons
which will collide with the molecules are sampled by the col-
lision probability. In each computational loop, only a fraction
of the electrons (the probability percent of all electrons) will
be considered to collide with molecules, and statistical errors
may be generated in the MCC process. Another reason is that
the coefficient of ionization and attachment and the mobility of
electrons used in the continuum model are fitted from the result
of energy solver. In the PIC-MCC model, these parameters
are taken as the properties of electrons and are computed
directly. The occurrence of ionization or attachment is deter-
mined based on the relationship between the kinetic energy of
electrons and the collision cross section directly. Nevertheless,
the close agreement between the prior continuum model and the
PIC-MCC model suggests that the local approximation used to

8. WANG et al.: HYBRID MODEL TO PREDICT ELECTRON AND ION DISTRIBUTIONS 427
Fig. 11. Distributions of the electron mean kinetic energy as a function of
the electric field computed from (solid line) the PIC-MCC and (dashed line)
the continuum models. Note that the higher electric field is located at the inner
electrode.
solve the Boltzmann equation in the prior continuum model is
valid.
The electron mean kinetic energy distribution in the negative
corona plasma is shown in Fig. 11. For the prior continuum
model, the electron mean kinetic energy distribution is exactly
the same for both positive and negative coronas (independent
of the discharge polarity) [18], [19]. The PIC-MCC model pre-
dicts the electron mean kinetic energy distribution pretty well
except that there are some fluctuations due to the nature of the
PIC-MCC computation. For both models, the electron mean
kinetic energy increases with the increasing electric field. Sim-
ilar to the continuum model, the electron mean kinetic energy
distribution from the PIC-MCC model varies from ∼2 eV at
the outer boundary to ∼10 eV at the discharge wire surface.
The mean energy obtained by the PIC-MCC model is slightly
greater than that predicted by the prior continuum model at the
plasma boundary where the field strength is low but is slightly
smaller than that by the prior continuum model near the wire
surface where the field strength is high. This phenomenon phys-
ically explains why the number density of electrons predicted
by the hybrid model is lower than that obtained by the prior
continuum model at the wire surface but higher at the plasma
boundary (Fig. 10).
IV. CONCLUSION
A hybrid scheme has been developed to predict distrib-
utions of electrons, positive ions, and negative ions in the
entire discharge gap for a negative wire discharge. A modi-
fied PIC-MCC model was used in the corona plasma region,
and a continuum model was used in the unipolar ion region.
Therefore, the hybrid approach combines the accuracy of the
kinetic model and the efficiency of the continuum model. The
computed voltage–current characteristic of the discharge agrees
well with that obtained by theoretical prediction (Cooperman’s
formula). The electron number density distribution and the
electron kinetic energy distribution in the corona plasma region
computed by the modified PIC-MCC model are similar to those
obtained from the prior continuum model, which suggests the
validity of the local approximation used to solve the Boltzmann
equation in the prior continuum model. Superior to the prior
continuum model which applies only to macroscopic discharges
with feature sizes much greater than the mfp of charge carriers,
the hybrid model is applicable to discharges ranging from
microscopic to macroscopic. The hybrid model can thus be
used to study corona discharges from nanostructures and across
microscale or nanoscale gaps.
ACKNOWLEDGMENT
The authors would like to thank anonymous reviewers for
their insightful comments and the access to the Minnesota
Supercomputing Institute at the University of Minnesota,
Minneapolis, and the High-Performance Computing Service at
the University of Wisconsin, Milwaukee.
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Pengxiang Wang received the B.E. and M.S. de-
grees in thermal power engineering from Harbin
Institute of Technology, Harbin, China, in 1994 and
2001, respectively. He is currently working toward
the Ph.D. degree in the Department of Mechanical
Engineering, University of Wisconsin, Milwaukee.
He is a Research Assistant with the Department of
Mechanical Engineering, University of Wisconsin.
His research interests include numerical simulation
of electrical discharges, microscale and nanoscale
plasma, and parallel computations.
Fa-gung Fan received the M.S. degree in mechanical engineering and the
Ph.D. degree in engineering science from Clarkson University, Potsdam, NY,
in 1989 and 1995, respectively. His graduate work was on vibration isolation
and aerosol dynamics.
He was a Member of Research and Technology Staff, Xerox Research Center
Webster, Xerox Corporation. He is currently a Principal Engineer with the Me-
chanical Engineering Sciences Laboratory, Fairport, NY. His research interests
include xerography and modeling and simulation of marking processes.
Francisco Zirilli received the B.S., M.S., and Ph.D.
degrees from Clarkson University, Potsdam, NY, in
1977, 1979, and 1985, respectively. His graduate
work focused on the development of finite-difference
and finite-element algorithms for the solution of free
and mixed convection problems.
He has been with the Xerox Research Center Web-
ster, Xerox Corporation, Webster, NY, for 26 years,
where he held positions in technology development
and product development and design and is currently
a Principal Engineer with the Global Development
Group. His primary responsibility is the use of computational models in the
areas of fluid dynamics and heat transfer applied to the development and
design of high-speed marking engines. His areas of interest include numerical
simulation of turbulent flows, conjugate heat transfer, free convection heat
transfer, and cooling of electronic components.
Junhong Chen (M’02) received the B.E. degree
in thermal engineering from Tongji University,
Shanghai, China, in 1995, and the M.S. and Ph.D.
degrees in mechanical engineering from the Univer-
sity of Minnesota, Minneapolis, in 2000 and 2002,
respectively. His graduate work focused on dc corona
plasmas and corona-enhanced chemical reactions.
From 2002 to 2003, he was a Postdoctoral Scholar
in chemical engineering with the California Institute
of Technology, Pasadena, where he worked on arc
plasma synthesis of nanoparticles. He is currently
a Professor with the Department of Mechanical Engineering, University of
Wisconsin, Milwaukee. His current research interests include corona dis-
charges, plasma reacting flows, and nanotechnology for sustainable energy and
environment.
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