PIC-MC Simulation of Charge Accumulation Process
Inside Teflon Film
R. Watanabe*, N. A. Gatsonis** and N. Tomita*
*Dept. of Mechanical Systems Eng., Musashi Institute of Technology, Tokyo, Japan
**Mechanical Engineering Department, Worcester Polytechnic Institute, MA, USA
Abstract. The charge accumulation process inside a Teflon film is numerically simulated with a Particle-in-Cell/Monte
Carlo methodology. Elastic and inelastic collision processes between irradiated electrons and Teflon are implemented
based on a model that combines theoretical and semi empirical approaches. The simulations predict the charge
accumulation process properly and provide predictions of the penetration depth inside the Teflon. Electron irradiation
experiments are also conducted for code validation. These experiment provide the transient charge distribution inside
dielectric materials such as Teflon, Kapton and PMMA and show the dependency of the penetration depth to the material
density. Improvements in the computational and experimental methodologies will enable proper code validation.
INTRODUCTION
Dielectric breakdowns and electrostatic discharges occurring on/inside spacecraft surfaces have been considered
as one of major causes of spacecraft malfunction and failures [1]. These concerns are heightened for GEO
(GEosynchronous Orbit) that can suffer from severe bombardment of high-energy electrons and other charged
particles. In order to avoid failures, detailed analyses of charging and discharging process inside surface materials
such as Teflon are essential. Most previous studies focused on surface charging in a low-energy plasma environment
and theorized that differential surface charging may result in catastrophic discharges [2]. Recently, however, it has
been pointed out that besides surface charging there is possibility that internal charging is also related to discharging
of spacecraft [3]. Although there are some practical estimations of discharge criteria based on empirical equations
[4], numerical simulations based on first principle are important to understand the phenomena. Clarifying the charge
accumulation process inside insulating polymers will assist in the analysis of spacecraft failures and allow the
prediction and prevention of dielectric breakdowns that might occur under severe electron irradiation in space.
In the present research, the charge accumulation processes inside a Teflon film are investigated by a combination
of computational and experimental methods. The simulation strategy is based on a combination of the Particle-inCell method and the Monte Carlo methodology to address collisional processes. Simulations are performed using a
modified two dimensional PIC-MC code PDP2 developed originally for bound plasma simulation [5]. Major
modifications include the Monte Carlo implementation of the elastic and inelastic scattering processes between
electrons and atoms consisting of Teflon (CF4) described by Palov et al [6]. Electron-phonon interaction and
trapping effect are also included in the estimation of total cross section. Simulation results of an incident electron
beam with energy of 20 keV are presented. In order to verify the computations, electron irradiation experiments are
conducted based on a measurement technique described by Watanabe et al [7]. Real-time measurements of the
charge distributions inside Teflon, Kapton and PMMA are also presented.
PHYSICAL MODEL
Collision processes between electrons and Teflon are modeled following Palov et al [6]. Electron energies are in
the range of 1 eV to 35 keV and the processes considered in the model are: (i) electron scattering by C or F atoms,
(ii) ionization of atoms, (iii) electron-phonon interaction and (iv) trapping. Palov et al derives these formulae from
CP663, Rarefied Gas Dynamics: 23rd International Symposium, edited by A. D. Ketsdever and E. P. Muntz
© 2003 American Institute of Physics 0-7354-0124-1/03/$20.00
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quantum electro dynamics and other experimental results. The cross sections of each process are calculated in
advance as a function of the electron energy.
(i) Elastic Electron Scattering
Electron scattering in Teflon is considered separately with Carbon (C) and Fluorine (F) atoms because the theory
of electron-molecule collision process has not been adequately developed. The mean free path for elastic collision
λ−el1 is calculated from
4πn ∞
(2l + 1)sin 2 δ l ,
(1)
k 2 l =0
where n is the number density of atoms in Teflon and k is the electron wave vector. The phase shifts δ l were
obtained by calculating the following differential equation,
2
dδ l
V (k , R)
=−
cos δ l ĵl (kR) − sin δ l n̂ l (kR) ,
(2)
dR
k
V (k , R) = Vstatic ( R) + Vexchange (k , R ) + Vcorr ( R) .
(3)
∑
λ−el1 =
where ĵl (kR) and n̂ l (kR) are spherical Bessel and Neumann functions of order l and R is the distance from the
center of the atomic core. Detailed descriptions for the potentials Vstatic ( R), Vexchange (k , R) and Vcorr (R ) are given in Ref.
[6].
The azimuthal ϕ and polar θ scattering angles are calculated based on the following double cross section
equation,
d 2 λ−el1
n
=
d (− cosθ )dϕ 2πk 2
∞
∑ (2l + 1) sin δ exp(iδ ) P (cosθ )
l
l
l
2
,
(4)
l =0
where Pl (cosθ ) is the l -th Legendre polynomial.
(ii) Ionization of atoms and generation of secondary electrons
The ionization of the inner atomic shells and a valence band are dominant as inelastic scattering processes
especially in a higher energy range. We used the classical theory of binary collision for the description of the
ionization of inner atomic shells and the generation of secondary electrons. The inelastic mean free path before
ionization of the j -th shell λi and electron energy losses are calculated from differential cross sections averaged
over kinetic energies of shell electrons,
2
1

dλi−1 πa0 nN j 
(1 − y / x )1+ y  y(1 − 1/ x ) + 4 ln 2.7 + x − y  x 3  ,
(5)
=
2* 3 
dy
U j y 
3

 (1 + x ) 
where x = E U j , y = ∆E U , E is the energy of a primary electron, ∆E is its energy loss, n is atomic
(
)
concentration, N j is the quantity of electrons on the j -th shell and U j (in atomic units) is the binding energy of the
j -th shell and a0 is the Bohr radius. The momentum loss ∆P of a primary electron is estimated by the following
double differential cross section for a fixed kinetic energy of a shell electron,
d 2 λi−1
πa02 nN −5 2
=
ξ
dy dξ 8 2 x E 2
ξ = 1 − y 2 x − 1 − y x cosθ , ∆P = 2 Eξ
(6)
E1 = E − ∆E , E s = ∆E − U j + χ
(7)
where E is in atomic units. E1 is the energy of the primary electron after collision and E 2 is the energy of the
secondary electron, where χ is the electron affinity of Teflon.
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Polar angles of the primary and secondary electrons are calculated from Eq. (6) and the energy and momentum
conservation laws. The azimuthal angle ϕ of the primary electron is uniformly distributed in the (0, 2ϕ ) interval,
the secondary one ϕ s is connected with ϕ by
cos(ϕ − ϕ s ) =
( p − c )( p cosφ − c )
t
(p
i
2
c = (2 p 2 − 2m∆E − ∆P 2 ) 2 p,
− c2 )
p = 2mE ,
t 2 = 2m∆E + p 2j − ( p + p j cos φ − c ) ,
p j = 2mU j ,
2
(8)
where φ is the angle between atomic and primary electron velocities before a collision and m is the electron mass.
(iii) Electron-Phonon Interaction
The electron-phonon interaction is also included in the model because it is important when electron energy is less
than the energy gap. We used Frohlich’s formulas [8] to estimate the mean free path of an electron before the
creation of a phonon λ−ph1 .
dλ−ph1
dq
=
(1 + n ) E
qmin =
2m
h
1 1
ph  1
 − 
E  ε ∞ ε 0  q
(
E − E ph
(
)
)
2m
E + E − E ph
(9)
h
is the energy of a fixed longitudinal optical phonon ε 0 and ε ∞ are the insulator dielectric functions in
qmax =
where E ph
q
2a 0
zero and high energy limits. q is the wave vector lost by an electron in the process of the creation of a longitudinal
phonon, nq = {exp(E ph KT ) − 1} , K is the Boltzmann constant and T is the temperature of the insulator.
−1
(iv) Polaronic effect and electron trapping
The polaronic effect is important because it describes electron trapping and detrapping inside insulators
especially when they have lower energy. We used a semi-empirical method [9] to calculate the mean free path
1
before trapping λ−trap
,
1
λ−trap
= a exp(− E b )
(10)
-1
where a =1 nm and b = 1.5 eV. The parameters a and b were adjusted according to the calculated TSEY with
experimental data.
COMPUTER MODEL
In order to model the motion and collisions of electrons inside a material, we used a PIC-MC strategy by
modifying the PDP2 code originally developed for bound plasma simulation [5]. Following PIC, a grid is used to
calculate the electric field induced by the accumulated electrons and created ions. The charge density on the grid is
obtained by a liner weighting of the neighboring particles and used to solve the electric field from Poisson’s
equation. A rectangular uniform mesh is used and Poisson’s equation is solved using the Dynamic Alternating
Direction Implicit (DADI) scheme. The electric field is reassigned to each particle and provides the force in the
particle mover, which utilizes the classical equations of motion to advance the particles to new positions and
velocities. A uniform time step ∆t is used for all computational particles. Then the collision cross sections are
estimated based on the physical model described above. The interaction, which will occur for each particle is chosen
randomly among the processes mentioned above. To reduce the computational cost, the cross sections are calculated
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10
1.5e-7 [s]
3.0e-7 [s]
4.5e-7 [s]
Charge Density [C/m 3 ]
5
Teflon
y
0
-5
-10
-15
x
-20
1 10 -5
0
2 10 -5
3 10 -5
4 10 -5
5 10 -5
X [m]
FIGURE 1. Simulated model system.
FIGURE 2(a). Charge density distribution.
in advance as a function of electron energy and the null-collision method is used in the Monte Carlo scattering
selection. The full three-dimensional character of a collision is modeled by utilizing three velocity components.
COMPUTED RESULTS
Figure 1 shows the physical system that represents the electron irradiation experiments to be described later.
Electrons are injected from the left side ( x = 0 ) which is grounded. The electron flux is J xL = 6.25 × 1022 m-2s-1
and the energy is Vel = 20 keV. The right side is connected to a coupling capacitor of 1.0 × 10-9 F. No external
voltage/current sources are considered here and periodic boundary conditions are applied to the top and the bottom
surfaces. Inside the domain, a Teflon film is placed which has a dimension of 50 µm × 50 µm in x and y direction
respectively. The density of Teflon is ρ = 2.1 g/cm3 and the relative permittivity is ε r = 2.2. We used 200 mesh
points in each direction resulting in grid spacing of 2.5 × 10-7 m. The time step is set ∆t = 5.0 × 10-13 s.
Figure 2 (a) shows the charge distributions inside the film at various times. Each distribution is obtained by
averaging over the sample in the y -direction. The computation lasts until the distribution seems to be saturated. It is
observed that the electrons accumulate at the depth of 20 µm although some noise is present owing to the nature of
PIC method. The penetration depth predicted by our simulation is about three times larger than Ref. [6]. Our results
show that there is accumulation of Carbon ions near the charge peak, which was not observed in the results of [6].
The difference may be attributed to the estimation of the cross sections or velocity assignment after collision in our
method. Figure 2(b) shows the electric field in x -direction calculated from the charge density on the grid points.
High electric fields are induced at the charge peak point and near the electrodes on the both sides. Figure 2(c) shows
the potential distributions with the negative peak to appear at the charge peak point.
ELECTRON IRRADIATION EXPERIMENTS ON DIELECTRIC FILMS
The experimental procedure and apparatus presented in [7] enables us to measure the charge distribution inside
dielectric materials in real time during irradiation. The Piezo-electric Induced Pressure Wave Propagation (PIPWP)
experimental procedure uses a pulsed pressure wave propagating in the sample as a charge probe [10]. When there
are electric charges in the sample, the position of chares move slightly by the pressure wave. The movement of the
charges induces the charge of surface charges on the electrode that causes displacement current in the external
circuit. The time-history of the displacement current indicates the charge distribution in the sample as described
below for various materials.
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1
6 10 5
1.5e-7 [s]
3.0e-7 [s]
4.5e-7 [s]
0
1.5e-7 [s]
3.0e-7 [s]
4.5e-7 [s]
-1
2 10 5
Potential [V]
Electric Field [V/m]
4 10 5
0
-2 10 5
-2
-3
-4 10 5
-4
-6 10 5
-5
0
1 10 -5
2 10 -5
3 10 -5
4 10 -5
5 10 -5
1 10 -5
0
X [m]
2 10 -5
3 10 -5
4 10 -5
5 10 -5
X [m]
FIGURE 2(B). Electric field distributions.
FIGURE 2(C). Electric potential distributions.
0.001
Electron Beam
0.0005
PMMA
[-30 s－5 min]
100
0
-0.0005
-0.001
-1 10 -4 -5 10 -5 0 10 0 5 10 -5 1 10 -4 1.5 10 -4 2 10 -4 2.5 10 -4 3 10 -4
Charge densityρ(z) [µC/cm3]
Charge density [C/m
3
]
125
75
315 µm
50
0 min
25
time
0
-25
-50
-33 µC/cm3
0
220 µm
5 min
z
510 µm
X [m]
FIGURE 3. Charge density distribution (Teflon)
FIGURE 4. Charge density distribution (PMMA)
Teflon Film
Figure 3 shows the charge density distribution inside a Teflon film with thickness of 190 µm. The incident
electron energy is 90 keV and the current density is 205 nA/cm2. The measurement was carried out after 10 minutes
irradiation. Two charge peaks are found at the electrodes, but there is no apparent charge accumulation inside the
film. The peak observed around 100 µm is not a significant signal. For the incident energy of 90 keV it was
expected that the penetration depth was several dozen µm. But the positional resolution of our current device is 40
µm. Thus, electron deposition within a range of the positional resolution from the irradiated surface cannot be
detected. The resolution can be improved by shortening the pulse width because it directly affects the detected signal.
Experiments using shorter pulse are underway and the resolution will become high enough to detect shallow
deposition.
PMMA Film
For lower density polymers, electrons are easy to penetrate deeply into the material. Figure 4 shows the charge
distribution in PMMA film with thickness of 510 µm. The incident electron energy is 230 keV and the current
density is 10 µA/cm2. In this experiment, the charge distribution is measured in real time for 5 minutes as seen in the
figure. Then you can see how the charges are accumulated in the sample. It is interesting to know that the charges
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start to accumulate evenly in the region between the irradiated surface and the maximum penetration depth (315 µm
in this case). Subsequently, they start to have a peak around 220 µm and reach an equilibrium state.
Kapton Film
electrode
3
charge density ρ(x) [C/m ]
Kapton is also widely used as an insulating material for spacecraft as well as Teflon. We conducted experiments
on Kapton films to see the difference of how charges are accumulated. Figure 5 shows the result of 75 keV electron
irradiation. In this case, the positive charge accumulation induced from created ions is observed as well as the
electron deposition at 65 µm.
Kapton
electrode
10
0
-10
0
80
160
position x [µm]
FIGURE 5 Charge density distribution (Kapton)
Summary of Experiments
The penetration depths of injected electron are summarized in Table 1. The densities of these materials are also
shown. Though the incident electron energies are not identical for all cases, it is obvious that the higher the density
is the more difficult to penetrate deeply, therefore, a more accurate measurement technique is needed.
TABLE 1. Electron Penetration Depths
Sample
Density (g/cm3)
Incident Energy (g/cm3)
Teflon (PTFE)
2.1
90
Kapton (polyimide)
1.41
75
PMMA
1.19
230
Penetration Depth (µm)
? (30-40?)
65
220
SUMMARY
A PIC-MC simulation of the charge accumulation process inside a Teflon film is carried out by a modified PDP2
code originally developed for plasma simulations. Major modification includes the Monte-Carlo implementation of
the collisional processes inside Teflon as described by Palov et. al [6]. Simulation results show the electron and ion
deposition processes. However, the predicted penetration depth is three times larger comparing with previous
simulations [6].
To aid in code validation an electron irradiation experiment on a Teflon film is conducted but no clear deposition
is detected inside the sample because of the insufficient positional resolution of our experimental system.
Experiments on Kapton and PMMA dielectric materials show that there is clear dependency of the penetration depth
to the material density.
Improvement of the computational model and the accuracy of the experimental method are needed to discuss the
charging characteristic in detail and provide means for proper code validation.
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ACKNOWLEDGMENTS
This study is carried out as a part of “Ground Research Announcement for Space Utilization” promoted by Japan
Space Forum and “Grant-in-Aid for Scientific Research (A)” promoted by Japan Society for the Promotion of
Science. We would like to thank Prof. Tanaka of Musahi IT for experimental operation.
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