The Effect of Charge Limits on Particle Charge Distributions
in Nanodusty Plasmas
Romain Le Picard and Steven L. Girshick
Dept. of Mechanical Engineering, University of Minnesota, Minneapolis, MN 55455, USA
Abstract: The effect of single particle charge limits on particle charge distributions in dusty
plasmas is examined. We show that charge limits can have significant effects on steadystate particle charge distributions and on the time required to reach steady state. Whether or
not charge limits have a significant effect is shown to depend on a dimensionless number
that characterizes the asymmetry between electron and ion currents to a particle.
Keywords: Particle charge limits, particle charge distributions, dusty plasmas.
1. Introduction
Evaluation of nanoparticle charging is necessary to
model the formation and growth of nanoparticles in dusty
plasmas [1]. Due to the high mobility of electrons compared to ions, dust particles tend to be unipolar negatively
charged. A variety of phenomena affecting particle charging, such as secondary electron emission, photoelectric
emission and ion trapping have been extensively studied
over the past couple of decades [2]. The average charge
on dust particles of given size is usually found by balancing electron and ion currents to the particle. As particle
diameter decreases, the discrete nature of charging, along
with charge fluctuation, becomes important [3] [4]. In
such situations, particle charge distributions, rather than
just average particle charge, become of interest. Plasma
parameters strongly affect charge distributions. Assuming
quasineutrality between electrons and ions in the bulk
plasma, i.e.
, Matsoukas et al. have shown numerically and analytically that stationary particle charge distributions are in most cases Gaussian [5]. However, in the
presence of a high-density cloud of nanoparticles, the
electron density can be strongly depleted, so that
quasineutrality occurs between electrons, ions, and
charged nanoparticles [6]. Under such circumstances the
electron-to-ion density ratio may be on the order of 10-1 to
10-3, which modifies particle charge distributions. Monte
Carlo simulations have shown that particle charge distributions deviate from Gaussian form at low electron-to-ion
density ratio [7].
Nanoparticles can hold a limited number of electrons—
referred to herein as their “charge limit”—due to the competition between electron affinity and inter-electron Coulomb repulsion. The existence of charge limits, along with
the coupling of dust particles with the plasma, motivated
us to study the effect of charge limits on particle charge
distributions. Different estimates of the magnitude of
charge limits have been derived. Draine and Sutin developed a criterion for charge limits based on electron field
emission for a perfectly conductive sphere [8]. Their result does not apply to nanoparticles made of semiconductor materials (e.g. silicon, germanium). More rigorous
expressions have been developed for silicon nanoparticles,
but ignoring inter-electron repulsion [9]. Gallagher derived an expression that considers this effect [10]. In that
work, the electron affinity (EA) of silicon as a function of
particle size was based on Fukuzawa et al., whose expression was derived by means of classical electromagnetic
theory [11]. In the current work, we modify Gallagher’s
charge limit formulation using more rigorous estimates
for the size-dependent EA of silicon and germanium nanoparticles obtained by Melkinov and Chelikowsky [12].
That work, which utilized ab initio pseudopotentials under
the local-density approximation, accounts for quantum
confinement in very small nanocrystals. Using this formulation, we study the effect of charge limits on particle
charge distributions.
2. Charge limits
Charge limits result from the competition between the
EA, inter-electron repulsion (δz), and particle surface potential (Vs). In the classical model, an electron will impact
a particle if its kinetic energy can overcome the surface
potential. In order to attach to the particle, the electron
needs to lose kinetic energy by collisions within the particle. Fig. 1 (left) shows the potential felt by an incoming
electron onto the particle of radius R. For radial locations
r > R the potential is given by the Coulomb potential.
Within the particle, the potential is assumed to be constant. The step in potential between the surface and the
interior of the particle equals the EA. When the electron’s
kinetic energy becomes less than the attractive potential
(inside the shaded region), it will be trapped by the particle. From Fig. 1 (right), we see that the shaded region is
reduced by the inter-electron repulsion, which raises the
potential within the particle. The EA is assumed to be
independent of particle charge. Therefore, the extent of
the shaded region decreases when the number of charges
increases, because of the higher surface potential.
Electrons will tunnel out of the particle when the shaded region disappears. This can be expressed by the condition:
(1)
Figure 1 Potential felt by an electron interacting with a
charged particle of radius R: Vs is the potential at the particle surface, EA the electron affinity and δz the interelectron repulsion. Left (right): without (with) considering
inter-electron repulsion.
The surface potential for a particle is classically defined
as
Figure 2 Charge limits obtained by Eq. (5) for silicon and
germanium nanoparticles.
(2)
where z is the total number of elementary charges, e is the
elementary charge and ε0 is the permittivity of free space.
The inter-electron repulsion term is evaluated by Gallagher [10] as
(3)
where ε is the material’s relative dielectric constant (ε ≈
12 for crystalline silicon and ~16 for germanium). The EA
of nanoparticles is smaller than the bulk EA (EAbulk) because of sphericity. As particles grow, their EA tends toward the EAbulk of the material. Here we use the EA for
silicon and germanium determined numerically by
Melnikov and Chelikowsky based on ab initio
pseudopotentials [12]. By using scaling parameters, they
give the general form for EA as:
3. Numerical model
We consider non-interacting nanoparticles in a plasma
of constant properties. Emission processes are neglected
(e.g. secondary electron emission). Ions are singly
charged. Electrons and ions do not collide with neutrals
around the dust particle, which is valid providing that the
mean free path is greater than the screening length.
Maxwellian energy distributions are assumed for electrons
and ions. Under these assumptions, electron and ion currents, Ie and Ii, to a particle can be described by the orbital
motion limited theory [13]. Within this model, electron
and ion currents depend on the number of charges z carried by a nanoparticle, and are determined by electron and
ion densities (ne, ni), temperatures (Te, Ti) and masses (me,
mi). For particles with surface potential Vs < 0:
(6)
and
(7)
(4)
while for Vs ≥ 0:
where aB is the Bohr radius, the bulk electron affinity for
silicon (germanium) equals 4.1 (4.4) eV, and scaling parameters are EA0 = 23.5 (30.8) eV and l = 0.9 (1.0). Using
Eqs. 1-4, we obtain the following formulation for the
charge limit:
(5)
To illustrate the effect of particle material on the charge
limit, Fig. 2 compares charge limits for silicon and germanium nanoparticles up to 100 nm in diameter. As expected, the charge limit is more severe for silicon since its
dielectric constant is smaller (i.e. stronger inter-electron
repulsion) and also its EA is less than for germanium. In
the rest of the paper, we focus on the effect of charge limits on particle charge distributions for silicon nanoparticles.
(8)
and
(9)
where I0e and I0i are the electron and ion currents at Vs = 0,
given by
for x ≡ e,i.
One can then solve the transient probability density
equation, which is formulated as a one-step Markov process:
(10)
where nz is the probability density for nanoparticles that
carry z charges. In this formulation, the number of charges
on a particle can only change by 1 with each discrete ion
or electron collection event. When the particle charge
limit is reached, no more electrons can attach to the particle, and therefore the electron current to the particle is
zero. Simulations start with a normalized neutral concentration. Eq. (10) is solved numerically in time until
steady-state is achieved to within specified tolerance. For
the results shown here, we take steady-state to be reached
when the relative error between two iterations is less than
10-10.
4. Effect of electron-to-ion density ratio
We first focus on the effect of electron-to-ion density
ratio on particle charge distributions, as the existence of a
nanoparticle cloud with negatively charged particles can
cause positive ion densities to sharply increase and electron densities to sharply decrease [6]. Typical values of
the density ratio are in the range 0.001-1. Charge distributions are solved using Eq. (10) until steady state is
reached. Calculations were carried out for monodisperse
50-nm-diameter particles, for ne/ni = 0.1, 0.03 and 0.01,
with and without the existence of charge limits. Other
simulation parameters are me/mi = 10-5 (as in argon plasmas) and Te/Ti = 100, which corresponds to electron and
ion temperatures of 2.6 eV and 300 K, respectively.
Results for normalized particle charge distributions are
plotted in logscale in Fig. 3. Charge distributions without
charge limit are shown by solid lines, while distributions
with charge limit are shown as histograms. These results
show that as the electron-to-ion density ratio decreases,
the effect of charge limits becomes small, and charge distributions with and without charge limit are almost the
same. The theoretical Gaussian distribution from Ref. [5],
shown at ne/ni=0.01, does not perfectly match the numerical results. This discrepancy at low values of ne/ni has
been observed in previous work [7]. Here, however, the
existence of charge limits is seen to invalidate the assumption of Gaussian charge distribution over a large
range of ne/ni.
5. Time to reach steady-state charge distributions
In this section, we focus on the characteristic charging
time, defined here as the time it takes for charge distributions to reach numerical steady state as described in section 3. Because of the much lower mobility of ions compared to electrons, the characteristic charging time is dominated by ion properties (mass, density and temperature).
It can be shown that charging time is inversely proportional to both ion density and dust particle radius [7].
To show charge limit effects, we solve Eq. (10) as in
section 4. Results are presented in Fig. 4. Charging times
for these particles are on the order of milliseconds. We see
that when ion density increases, reaching steady state
takes much less time. We also see that the existence of
charge limits significantly reduces the charging time, provided that the electron-to-ion density ratio is not too low.
Figure 4 Charging time with and without charge limits for
50-nm-diameter Si particles for various values of ion density (cm-3) at Te/Ti = 100 and me/mi = 10-5.
6. Criterion for whether effect of charge limits is significant
When the electron-to-ion density ratio is sufficiently
small, charge limit effects become small, as suggested in
Fig. 3 for ne/ni = 0.01. In this section, we present a criterion for determining whether the effect of charge limits is
significant, based on the dimensionless number p, defined
as
(11)
Figure 3 Charge distributions for 50-nm-diameter Si
particles without charge limit (solid line) and with
charge limit (histogram), for various values of ne/ni.
which characterizes the asymmetry between electron and
ion currents to a particle. For a given particle size, we
numerically solved for charge distributions with and
without the existence of charge limits until steady state is
reached for various values of p. We consider the effect of
charge limits to be negligible when
(12)
We repeated these calculations for a range of particle
diameters over the range 10-100 nm, and obtained the
following empirical correlation:
(13)
where R is the particle radius (in nm) and pneglect is the
value of p below which the existence of charge limits has
negligible effect on steady-state charge distributions. For
cases of high dust density, the electron-to-ion density ratio
varies more significantly than do temperature and mass
ratios. Therefore, we show in Fig. 5 values of ne/ni for
which the effect of charge limits on stationary charge distributions can be neglected for two different electron temperatures. We see that the effect of charge limits becomes
more important as both electron temperature and particle
size increase. This can be explained by noting that both
increasing electron temperature and particle size cause
particles to be more negatively charged, so that the existence of charge limits poses a greater constraint compared
to the case without charge limits. In both cases, reducing
the value of ne/ni tends to relieve this constraint.
The criterion given by Eq. (13) may change by extending the analysis to non-Maxwellian plasmas, and by including other charging processes such as secondary electron emission. We leave such analyses to future work.
Figure 5 Value of electron-to-ion density ratio below
which the effect of charge limits on stationary charge
distributions can be neglected, for two different values
of Te. Plasma parameters are me/mi = 10-5 and Ti = 300
K.
7. Conclusion
In this paper, we formulated a rigorous expression for
the material-dependent limit on the negative charge that
can be carried by a nanoparticle in a plasma. A comparison was made between the charge limit for silicon and
germanium nanoparticles. Due to its smaller electron affinity and dielectric constant, the charge limit is more severe for silicon. The electron-to-ion density ratio is the
main factor that determines the extent to which the existence of charge limits affects stationary charge distributions. Providing that this ratio is not too low, the effect on
particle charge distributions is significant. We showed
that the existence of charge limits can reduce the characteristic particle charging time by up to an order of magnitude as the electron-to-ion ratio becomes close to unity.
8. Acknowledgments
This work was partially supported by the US National
Science Foundation (grant CHE-1124752), US Department of Energy Office of Fusion Energy Science (grant
DE-SC0001939), and the Minnesota Supercomputing
Institute.
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