Paper No 7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
Dancing on an Electric Curtain
Guanqing Liu1 and Jeffrey S. Marshall2
'Tsinghua University, Department of Thermal Engineering
Beijing, 100084, China
2University of Vermont, School of Engineering
Burlington, VT 05405, U.S.A.
jeffm @cems.uvm.edu
Keywords: electric curtain, discrete element method, electric field
Abstract
A hybrid discreteelement / boundaryelement method has been developed for simulation of adhesive particle transport by
traveling and standing waves on an electric curtain. The study shows that both wall adhesion and particleparticle collisions
have an important influence on particle transport by traveling waves at different wave frequencies. The most significant effect
of particle collisions on transport in traveling waves occurs with medium frequencies, in which particles with large negative
charge collect in highconcentration bands and move in a synchronous surfing mode, pushing forward particles with lower
charge. Cases with higher and lower frequencies exhibit hopping motion, for which adhesion determines the range of
nontransported particles. It is also demonstrated that standing waves in the electrostatic field can produce a nonzero net
transport of the particles through two distinct modes, one involving continuous rolling of particles along the surface and one
involving a random hopping motion of the particles.
Introduction
Electric curtains consist of a series of parallel electrodes
covered by a dielectric surface, across which is propagated
lowfrequency traveling or standing waves in the electric
field. Electric curtains are known to both transport and
confine dust particles, depending on the operating
conditions. They have been used for separation and sorting
of particles of different sizes and charges (Kawamoto, 2008;
Masuda et al., 1987; Machowski and Balachandran, 1997)
and for classification of particle size (Kawamoto and
Hasegawa, 2004). The primary applications of electric
curtain technology are dust mitigation for space exploration
on dusty planets and moons (e.g., Mars and the Earth's
moon) (Atten et al., 2009; Calle et al., 2008, 2009) and
conveyance of xerographic toner particles in photocopying
machines (Melcher et al., 1989; Schmidlin, 1991). Other
proposed applications include aerosol cloud confinement
(Masuda et al., 1972), blood cell sorting and transport
(Masuda et al., 1988), radioactive dust control in fusion
reactors (Onozuka et al., 1997), agricultural seed sorting
(Weiss and Thibodeaux, 1984), bubble transport and control
(Aoyama et al., 1993), and liquid droplet manipulation
(Kawamoto and Hayashi, 2006).
Three distinct modes of particle motion on an electric
curtain have been identified (Melcher et al., 1989b;
Schmidlin, 1995), which are referred to as the surfing mode
(SM), the hopping mode (HM), and the curtain mode (CM).
Various factors that affect the mode of motion include the
wave frequency, waveform and voltage, the electrode
geometry and configuration, the particle physical properties
(such as size, charge, surface energy), and the particle initial
position. These three modes are illustrated schematically in
Figure 1. Under surfing mode (Figure la), particles roll or
slide on the plate surface in a synchronous motion (i.e., at an
average velocity equal to the wave speed), with an
instantaneous velocity that varies between about 10% above
and about 10% below the wave speed. Particles in hopping
mode (Figure lb) are transported through successive hops,
inbetween which particles remain fixed to the surface. The
traveling distance of each hop can vary depending upon
specific conditions as well as between different hops of the
particle, even for synchronous HM. In some cases a particle
may perform several small hops, or even a backwards hop,
before resuming large forward hops. In curtain mode
(Figure Ic), particles are continuously suspended above the
plate and exhibit a periodic cycloidic motion with a drift
velocity much smaller than the wave speed. In practice, the
motion of an individual particle is usually more complicated
and irregular than is described by these three modes of
motion, in part due to the effects of interaction between
particles (Gartstein and Shaw, 1999; Liu and Marshall,
2010a). Nevertheless, these idealized modes provide a
framework with which to describe aspects of the motion
observed in manyparticle systems.
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
(b)
dielectric surface
(a)
y
D L
(b)
0a VW W
(C)
Figure 1: Schematic diagram showing (a) particles rolling
or sliding along the curtain in surfing mode (SM), (b)
particles intermittently jumping along the curtain in
hopping mode (HM), and (c) particles spirally above the
electric curtain in curtain mode (CM).
The current paper examines the effects of particle adhesion
in a manyparticle systems with both traveling and standing
wave fields on an electric curtain. In order to accurately
capture the effects of particle adhesion, a discreteelement
model (DEM) with van der Waals adhesive force is
employed (Liu et al., 2010). This method is based on
extending a recently developed discreteelement method for
adhesive particles by Marshall (2009) to account for electric
field forces and electrostatic interaction of particles. We use
this numerical method to study the dynamics of particle
transport and mutual interaction under imposed traveling
and standing waves with different oscillation frequencies.
Nomenclature
D electrode width (m)
E electric field vector (Vm 1)
F force (N)
H electrode embedded depth (m)
L electrode separation (m)
Q particle charge (C)
S dimensionless net travel distance
a, b fitting coefficients
c wave speed (ms1)
d particle radius (m)
f frequency (Hz)
p dipole moment (Cm)
q, q' dimensionless particle charge
r particle radius (m)
t dimensionless time
Wi average transport velocity
y average levitation height
Subscripts
d dielectric material (curtain surface)
c, C critical
f fluid
p particle
P peak
Greek letters
D adhesion parameter
electrode voltage (V)
Figure 2: Electrode geometry for (a) standing waves and (b)
traveling waves on the electric curtain.
e permittivity (Fm 1)
y surface energy (Nm 1)
A wavelength (m)
p fluid viscosity (kgm 's1)
p density (kgm 3)
Numerical Scheme
The electric field is generated by a periodic array of
electrodes embedded within a dielectric material with
permittivity sd The electrodes have infinite length, and
their centroids are separated from each other by a distance L
and from the top surface of the dielectric material by a
distance H. Electrodes of circular crosssection with
diameter D are used for computations with standing waves
(Figure 2a) and linear electrodes of length D for the
computations with traveling waves (Figure 2b). The electric
field is assumed to be periodic in the two horizontal
directions. For traveling waves, four electrodes are located
within the computational domain, with phases separated by
900. For standing waves, two electrodes are located in the
computational domain with phases 1800 apart.
A discreteelement approach similar to that of Marshall
(2009) is used to model particle collision, adhesion and
fluid forces. The wellknown JohnsonKendallRoberts
(1971) (JKR) model is used to model normal particle
collision force in the presence of van der Waals adhesion.
Particle rolling resistance in the presence of adhesion is
modeled using the expression by Dominik and Tielens
(1995). Both sliding and twisting resistance are included in
the model, with modifications to account for adhesion
(Marshall, 2009). The fluid is assumed to be stationary, and
so the fluidinduced forces include drag and Magnus lift
force, together with the viscous torque imposed by air
resistance on a spinning particle.
The electric field induced by the particles is modeled by
approximating each particle as the combination of a point
charge and a point dipole. The particleinduced electric
fields are computed using an optimized multiple
acceleration method (Liu et al., 2010), both for particles
within the computational domain and those in one period
adjacent to all sides of the computational domain. Particle
images over the dielectric surface are imposed using the
analytic solution of Sometani (2000). The electric field
induced by the electrodes and the dielectric material is
computed by decomposing the imposed AC potential in
time and then employing a twodimensional
boundaryelement solution approach (Liu and Marshall,
2010a). Each particle is subjected to a Coulomb force
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F, = Q E and a dielectrophoretic force F, = p VE where
the dipole moment p is proportional to the electric field
vector E.
The electric field generated by the electrodes is computed
with a twodimensional electrostatic boundary element
method (BEM). The BEM computation is applied with five
neighboring periods on each side of the computational
domain included in the computations. The electrodes and
dielectric surface within the computational domain are
discretized using 200 (400) evenly distributed line elements
on each electrode for the electric curtain with standing wave
(traveling wave) and 800 line elements on the dielectric
plane boundary. Following inversion of the
boundaryintegral equations to obtain the surface sheet
charge density on the electrodes and dielectric surface, the
electric field is computed on a uniform twodimensional
space grid covering the xy plane. In DEM simulations, the
precomputed electric field is interpolated onto the particle
centroids to calculate the Coulomb and dielectrophoretic
forces exerted by the curtain.
All DEM computations are performed in three dimensions
using 800 particles with diameter d/2= 0.0125 for
standing waves and 1250 particles with diameter
d/1 = 0.00625 for traveling waves. These choices are
sufficient to provide a single but dense particle layer on the
dielectric surface. At the beginning of the DEM simulations,
particles are initialized by first distributing the particles on
the curtain surface in a uniform array and then introducing a
random position perturbation of each particle in the xz
plane.
It is useful to express the results of the numerical analysis in
dimensionless form, using the characteristic scaling for
particle charge, velocity and length. In the following, we
normalize particle charge by a fixed nominal charge qo,
variables with dimension of length by the wavelength 2,
and time by the inverse wave frequency 1/f A
dimensionless particle charge q velocity components u
and v, and positions x and y are nondimensionalized using
this scaling, where in accord with the above velocity is
nondimensionalized by the wave speed c = f The
adhesion force between particles and the dielectric surface
is characterized by the critical adhesion force F, = 3rfyr .
To measure the relative importance of adhesion force to the
electrostatic force, a dimensionless adhesion parameter q
is defined as
37Lyrr
) = yr (1)
Eoqo
The values of all of the parameters used in the current
computations are listed in Table 1.
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
Table 1: Parameters used in numerical computations.
Computational
Parameter Values Unit
Traveling Standing
Waves Waves
particle radius, r, 25 25 pm
particle density, 3520 3000 kg/m3
P,
particle relative 3 3
permittivity,
1p/ ef
fluid viscosity, p 1.8x105 1.8x105 kg/ms
fluid density, pf 1.2 1.2 kg/m3
nominal charge, 105 105 elementary
q0 charges
particle charge
mean value 8.53 3 to 3 fC
standard deviation 9.22 fC
electrode width, D 1 1 mm
electrode 2 2 mm
separation, L
electrode 40 550 pm
embedded depth,
H
wavelength, .2 8 4 mm
wave frequency,f 10 200 10 100 Hz
electrode voltage
(peaktopeak), 1600 3200 V
Opp
relative
permittivity of 1.3 5
dielectric material,
ed l/f
computational 8x16x4 4x32x2 mm
domainXxYxZ
simulated 2.4 4.0 s
transport time
Results and Discussion
Traveling Waves
This section examines the effect of particle adhesive force
and wave frequency on the distribution of the particle
velocity and height. Cases are examined with all
combinations of the low, medium and h(gt dimensionless
frequencies (Cases FL, FM, FH) and adhesion parameter
values (Cases AL, AM and AH) listed in Table 2. All
computations in this section are performed with a Gaussian
charge distribution with dimensionless mean and standard
deviation values of 0.533 and 0.576, respectively. Particle
transport characteristics are evaluated based on modes of
motion and variation of timeaveraged transport velocity
and levitation height with particle charge.
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
Table 2: Parameter settings for simulation cases, with
values for dimensionless frequency (and corresponding
frequency in example calculation) and adhesion parameter.
Case Parameter value
FL (low frequency) f'=0.0264 (f=10Hz)
FM (medium frequency) f' = 0.0791 (f = 30 Hz)
FH (high frequency) f' = 0.132 (f = 50 Hz)
AL (low adhesion) F = 0.042
AM (medium adhesion) F = 0.335
AH (high adhesion) F = 0.628
12$  r ~ r  "r
0.75
as 0 FL
o
FM
0.25
I I I
0.f5 I
0o I I I FH
2 0 I I 1
q q'
(a)AL
(b)AM
(c) AH
Figure 3: Profiles of timeaveraged velocity versus
particle charge for different combinations of the
values of frequency (rows) and adhesion parameter
(columns) given in Table 2. Particles are sorted into 25
bins according to charge level, and a symbol is plotted
for the mean velocity in each bin and error bars are
plotted for the rootmeansquare value in each bin.
c; q q
(a)AL
(b) AM
(c) AH
Figure 4: Profiles of timeaveraged levitation height
versus particle charge for the same cases considered in
Figure 3
Figure 3 shows the timeaveraged transport velocity u
distribution as a function of particle dimensionless charge
q' for the nine combinations of the three different
frequency and adhesion parameter levels listed in Table 2.
The corresponding plots of timeaveraged levitation height
y for the same nine cases are given in Figure 4. A
comparison of the distributions for these nine cases contains
a wealth of information. Starting with the u distribution
profile for the case FLAL, there appears to be two states:
one for slow transport with i a little above zero and the
other for synchronous transport with uW 1 Highly
charged particles (HCP), regardless of the sign of charge,
have higher transport velocity. The u versus q' curve
shows that there exists a threshold charge q' such that for
Iq' > q' particles are moving synchronously with the wave
and for Wq' < q' transport velocity reduces as a function of
particle charge. Observation of particle motion reveals that
particles in the group with Wi 1 are undergoing hopping
mode. The transport distance in the xdirection varies from
onequarter of the wavelength for particles with relatively
low charge to onehalf or threequarters of the wavelength
for particles with higher charge. The hop distance is also
dependent on the relative position of the particle to the
electrode that launches it. Particles launched with a lower
angle with respect to the plate usually have a smaller hop
distance and a lower maximum levitation height. Particles
with low transport velocity (and low charge) in the FLAL
case move intermittently, sticking to the dielectric surface
inbetween motions. A large portion of lowcharge particles
concentrate on the edges of the electrodes, where the
electric field gradient is relatively high, leading to strong
attractive dielectrophoretic force. The timeaveraged
levitation height y increases roughly linearly with the
absolute value of the particle charge, as indicated by the
FLAL case in Figure 4.
As adhesive force is increased in cases FLAM and FLAH,
the results in Figure 3 maintain a roughly similar shape but
the region of particle charges corresponding to small
transport velocity becomes broader, corresponding to an
increase in the value of the threshold charge q' The
average levitation height maintains approximately the same
as for the AL case for the highcharge particles, but
decreases substantially with increase in adhesion parameter
for the lowercharge particles. Comparing the top rows
(Case FL) in Figures 3 and 4, it is apparent that particles
with low transport velocity also have approximately zero
levitation height, supporting the conclusion that these
particles are stuck to the dielectric surface.
Significant differences are observed between computations
performed with the low and medium frequency values. First,
it is interesting to find that in the FMAL and FMAM cases,
particles with few charges are also being effectively
transported, with velocity higher than about 20% of the
wave speed. For the low frequency (FL) case, by contrast, a
substantial number of lowcharge particles possess
nearzero transport velocity, especially for at the medium
adhesion (AM) level. Second, the symmetry in the u
versus q' curve which is apparent in the low frequency
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cases is now broken, and we instead observe substantially
reduced transport velocity for positively charged particles
compared to particles with the same magnitude charge of a
negative sign. Repetition of these computations without
particle collisions accounted for exhibit neither of these two
features.
The asymmetry in the FMAL and FMAM cases is caused
by the 'sweeping' effect of particle collisions as a band of
tightlyspaced particles moves along the plate in surfing
mode. Observation of particle motion shows that the
particles collect together as a tightlypacked band
positioned over an electrode during part of the wave cycle,
and then rapidly move forward along the dielectric surface
as a band during the other part of the wave cycle. This
highspeed band collides with lowcharge particles lying on
the dielectric surface and carries them along for a certain
distance. This mechanism allows particles with almost no
charge to be effectively transported. However, lowcharge
particles eventually work their way through the particle
band, due in part to strong resistance by wall friction and
adhesion, so that after a short time these lowcharge
particles are left behind and wait for the passage of the next
particle band.
The effect of adhesion is slight for the moderate frequency
cases with adhesion levels AL and AM. A more significant
difference is observed for the case with the highest adhesion
level (AH), where synchronous motion is absent and
negativesign highcharge particles have a much higher
levitation height compared to the AL and AM cases. The y
and iT profiles for the FMAH case is qualitatively similar
to those for the lower frequency computations, with low
transport velocity for lowcharge particles and a nearly
symmetric velocity profile. Due to the high adhesive
resistance, many particles stick to the dielectric surface.
There still exists a particle band moving in the wave
direction for the high adhesion case, but it is asynchronous
and contains many fewer particles than observed for
computations with lower adhesion levels. The particle
motion seems to intermittently alternate between surfing
mode, hopping mode and sticking on the surface. The
differences in particle motion observed between the high
and low adhesion parameter levels reveals the important
role that adhesion force plays in the transition from surfing
mode to hopping mode (Schmidlin, 1995).
For the highest frequency computations (FH), there are no
particles that exhibit synchronous motion due to the high
wave speed. The particle velocities are much smaller than
the wave speed and exhibit both forward and backward
velocities. These backward velocities arise due to
higherorder modes in the electric field (Masuda and
Kamimura, 1975). The velocity profiles for the FH
computations are also more scattered compared to lower
frequencies, where particles with similar charge might
either undergo forward or backward motion, as indicated by
the error bars in Figure 3. Two small local minima are
observed for the transport velocity profiles for cases AL and
AM, both with negative velocity, and the maximum velocity
magnitudes within these regions tend toward zero as the
adhesion is increased. The fastest particles move in an
asynchronous hopping mode, with forwardmoving
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
particles having a higher average levitation height than
backwardmoving particles. The backwardmoving
particles are also observed to move along the dielectric
surface intermittently before hops. The levitation height
increases nearly linearly with amount of charge and the
slope of the y versus q' curve remains almost unchanged
for the three adhesion levels, although the range of the
linear part of curve is shortened and the nonlevitated region
for lowcharge particles is broadened when the adhesion
force is strengthened. Similar to what was observed at low
frequencies (but contrary to the medium frequency cases),
the u and 7 profiles are nearly symmetric with regard to
particle charge, indicating low incidence of collisions
between particles of opposite charge.
Standing Waves
In a standing wave, the electric fields lines generated at one
electrode terminate on one of the neighboring electrodes, as
shown in Figure 5. For this reason, it has long been the
conventional wisdom in the electric curtain literature that
standing waves produce no net particle transport, but
instead the particles simply oscillate between the electrodes
(Masuda et al., 1972; Dudzicz, 1989). Experimental studies
with standing wave, on the other hand, appear to give
different results. For instance, Hemstreet (1985)
commented upon the existence of two modes of particles in
standing wave experiments on an electric curtain. One
mode was levitated above the surface and oscillated back
and forth as observed in previous experiments, but a second
mode of particles at higher elevations was observed to be
transported at high velocities (in both directions) by the
standing wave field. Experiments on dust mitigation
systems for space applications by Sims et al. (2003) and
Atten et al. (2009) further demonstrated the existence of
active dust transport by standing wave excitation of electric
curtains. In the current computations, in addition to trapped
particles oscillating between each pair of electrodes, we
observe two distinct modes of particle net transport, which
we refer to as surfing mode and hopping mode transport.
The presence of particle collisions is found to interfere with
the surfing mode transport with sufficiently high particle
concentrations and broad charge distributions.
0 0.25
x
Figure 5: Contours of the potential field, and
associated electric field lines, inbetween electrodes for
standing wave excitation of the electric curtain.
Paper No
B 8 0 B
A At At
,.,2
Figure 6: (a) Schematic showing motion and positions
of trapped levitated particles in different regions and (b)
closeup plot showing average levitation heights of
particles in corresponding regions.
Trapped Particles
Different modes of trapped particle motion exist on the
electric curtain, occurring primarily for cases with higher
oscillation frequency and lower particle charges. As shown
in Figure 6, three distinct regions of trapped particles are
observed. Particles with very low charge adhere to the
dielectric surface with negligible motion, whereas some
particles with slightly higher charge also lie on the surface
but roll back and forth in response to the electric field
variations (Mode C). Particles with medium charges can
also be trapped as they are levitated above the surface.
Mode B consists of particles that are continuously levitated
and oscillate approximately horizontally as the electrode
voltage is oscillated. By contrast, particles in Mode A hop
vertically up and down above each electrode. It is observed
that particles in Mode B seem to be more stable than those
in Mode A, particularly as the charge level increases. The
simulations indicate that certain particles that are initially
trapped in Mode A slowly shift in horizontal direction,
moving into Mode B at later time.
Surfing Mode Transport
In surfing mode motion, particles roll over the dielectric
surface in a continuous direction. In Figure 7 is plotted the
particle average velocity (top row) and average levitation
height (bottom row) as a function of charge for different
oscillation frequencies. The results in Figure 7 are
computed with no particleparticle collisions, and a similar
plot is shown in Figure 8 which includes particleparticle
collisions. A prominent feature of Figure 7 is the existence
of certain bands in the charge distribution in which the
particles move at the wave speed in both the forward and
backward directions. The average levitation height of the
particles within these bands is nearly zero, indicating that
the particles are rolling along the surface in surfing mode.
These surfing mode bands become wider with increase in
excitation frequency. For particles with charge less than the
minimum charge for the surfing mode band, both the
particle velocity and levitation height are nearly zero,
indicating that the particles remain stuck to the surface or
are trapped between the electrodes (as illustrated by Mode C
in Figure 6).
When particle collisions are included in the calculation,
these surfing mode bands disappear for cases with a broad
particle charge distribution (Figure 8). Other calculations
were performed with a narrower charge distribution,
spanning only the charge values within the surfing mode
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
band. These calculations do exhibit the surfing mode
motion, even with particle collisions included; however, it
is observed that after some time the particles stop
propagating in both left and right directions (which leads to
large numbers of collisions) and will instead begin to all
propagate in one direction, either toward the left or toward
the right. This is thus an interesting example of emergent
behavior, in which particle interactions via collisions
changes the essential characteristics of the particle transport,
leading to selection of a different manner of particle
transport.
The average speed of particle motion in surfing motion is
such that they traverse one electrode spacing distance (L)
for each halfperiod of electrode oscillation. To illustrate
this motion, we consider a positive charged particle which
at the start of the motion is located near a positive electrode
(Figure 9). In the region to the right of the positive electrode
and to the left of the next negative electrode (labeled 'R' in
Figure 9), the positivelycharged particle is attracted to the
right by the Coulomb force. The particle momentum is such
that it overshoots the next (negative) electrode and moves
into the region to the right of the negative electrode (labeled
'L' in Figure 9), in which the negative electrode exerts a
force pulling the particle back to the left. At this point, the
electrode charge changes sign, such that the positive
electrodes become negative and vice versa. The particle
then finds itself to the right of a positivelycharge electrode
and the process repeats itself. In this way the particle can
continue rolling in one direction on the dielectric surface
even for the case of a standing wave in the electric field.
' 4 V _
(a) (b) (c) (d)
Figure 7: The variation of timeaveraged transport
velocity i (top row) and levitation height y (bottom
row) versus particle charge q, at frequencies (a) 25 Hz,
(b) 50 Hz, (c) 75 Hz and (d) 100 Hz. The computations
neglect particleparticle collisions.
i
q q q q
(a) (b) (c) (d)
Figure 8: Same cases as in Figure 7, but this time
including particleparticle collisions.
Paper No
RR6
1 P2 P3
la~
O0
S
Figure 9: Schematic illustrating minimum particle
travel distance during a halfperiod of the electric field
oscillation.
Hopping Mode Transport
The scattered region in Figures 7 and 8 corresponding to
particle charge values higher than the surfing mode band
limit indicates particles that exhibit hopping motion. The
dependence of iW on particle charge for particles in this
region is rather weak. A slight increase of i with I q I is
observed for f = 25 Hz; however, there appears to be no
discernable dependence for f = 50 Hz. The average
levitation height y on the other hand, varies almost
linearly with q for high charge values. The small increase of
u with I q would appear therefore to be partially due to
the increase of the particle height with I q as the
electrostatic force decreases with elevation. This trend is
even more pronounced at lower frequencies. For instance,
in Figure 10 is shown plots for average particle velocity and
levitation height for f = 10 Hz. The average velocity
values (Figure 10a) scatter in a bowtie shape, with
increasing magnitude as the particle charge increases. Some
cases are observed with average particle velocity of up to 3
times the wave speed. The maximum instantaneous particle
velocities plotted in the x and ydirections (Figures 10cd)
indicate values that can reach 300400 times the wave
velocity during the hopping motion. The observation that
some particles move under standing wave excitation at high
velocities and high elevation distances above the dielectric
surface is consistent with the experimental observations of
Hemstreet (1985).
Particles with high electric mobility are attracted to the
electrodes and have time to gather near to the electrodes
during the oscillation cycle. When the electrode charge
changes, these particles receive primarily vertical impulse
by the Coulomb force, causing the particles to be elevated
off of the dielectric surface and leading to a hoppingtype
particle motion. These particles can rise to heights above the
surface many times the wavelength, and the hop length can
similarly exceed several multiples of wavelength. The
particle motion in this mode is highly irregular, with
particles changing directions and hop length in a seemingly
random fashion. The macroscopic transport behavior in the
hopping mode can be examined by plotting the net travel
distance S(t) averaged over a group of particles versus
time. The computed data is fit using a curve of the form
S(t) = atb
Values of the a and b coefficients for different cases are
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
recorded in Table 3, both for computations with and without
particle interactions included. The observed bestfit values
for the exponent b are very close to 12, which corresponds to
a onedimensional Brownian motion (Russel et a., 1989).
The similarity between the observed values of the spreading
coefficient b and the theoretical value suggests a similarity
between the hopping mode transport and a random
Brownian process. The values of the a coefficient are
observed to decrease by 2030% with the inclusion of
particle interactions, resulting in a lower overall particle
transport velocity. A comparison of the data from Table 3
with Brownian motion is given in the loglog plot in Figure
11, in which lines are plotted with slope 0.5 representative
of Brownian motion for each of the four
(b) q (d)
Figure 10: Plots for the low frequency (f = 10 Hz)
case showing (a) average particle velocities, (b)
average levitation height, and maximum instantaneous
particle velocities in the (c) xdirection and (d)
ydirection, all as a function of particle charge.
Table 3: Listing of bestfit coefficients a and b for
curves of the form S(t)= atb between net distance
traveled by a particle and time for particles in hopping
mode transport with I q 1>1.5.
Frequency Particle Mutual a b
(Hz) Interactions?
25 No 1.14 0.456
25 Yes 0.768 0.493
50 No 0.727 0.521
50 Yes 0.569 0.542
Paper No
14
12
10 
8 '
S(t) 6 4
,. .
4
10 100 200
t(T)
Figure 11: Variation of net transport distance of a
group of hopping particles versus time, where particles
have charge q > 1.5. Computations are performed forf
= 25 Hz (squares) andf= 50 Hz (triangles), both with
particle interactions (open symbols) and without
interactions (filled symbols). Lines are fit with the
theoretical slope 0.5, with dashed lines for the cases
with particles interactions and solid lines for the cases
without interactions.
data sets. It is apparent from this plot that Brownian motion
is a reasonable approximation to the particle transport under
hopping mode, but that inclusion of collisions acts to
shorten the particle meanfreepath and hence reduces the
rate of particle spreading.
Conclusions
Adhesive particle transport by traveling waves on an
electric curtain is studied using a hybrid softsphere
discreteelement / boundaryelement method. Particle
transport behavior is investigated under different
frequencies and adhesion levels for both traveling and
standing waves on the curtain.
For traveling waves, the predominant mode of particle
motion at relatively low adhesion levels changes from
hopping mode at low frequency to surfing mode at medium
frequency, degenerating into an irregular motion at higher
frequency. Particle transport is found to be especially
effected by collisions at medium frequency levels, for
which particles with relatively high charges move in a
synchronous surfing mode on the dielectric surface in
highconcentration bands. These particle bands sweep
forward other particles with low charges, which acts to
provide a net forward transport of particles with low or zero
charge. At high frequency, there exists substantial number
of particles transported in both forward and backward
directions. The forwardmoving particles have higher
charge magnitude than do the backwardmoving particles.
For standing waves, two distinct modes of net particle
transport are observed, corresponding to surfing mode and a
hopping mode motions. In the surfing mode, particles roll
along the surface in a single direction at average velocities
equal to the wave speed. Surfing mode transport occurs only
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
within discrete bands of particle charge, where both the
band width and the limiting charge values increase with
increase in oscillation frequency. Hopping mode transport
occurs at high values of particle charge, in which particles
spread outward in a Brownianlike stochastic process. The
individual particles can travel in a single hop distances of
many times the electrode separation distance and be
characterized by velocities much larger than the wave speed.
The average net distance of particle travel due to the
hopping motion varies approximately with the square root
of time, similar to the case of Brownian motion. The
hopping motion transport is particularly prevalent for low
frequency cases, whereas at high frequency the particles not
undergoing surfing mode transport are generally levitated
above the plate and trapped within a single cell. It is
observed in computation involving both hopping and
surfing mode particles that particle collisions can disturb the
surfing mode, forcing all particles to enter into a hopping
mode regime. For narrow particle charge distributions
centered around the surfing mode bands, particles are
observed to exhibit surfing mode behavior even in the
presence of collisions, but the particles over time select a
single direction of motion so as to minimize frequency of
particle collisions.
Acknowledgements
This work was supported by NASA under cooperative
agreement NNX08AZ07A (Dr. Carlos Calle program
manager). G.Q. Liu thanks the China Scholarship Council
Postgraduate Scholarship Program for financial assistance
during his jointtraining visit to the University of Vermont.
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