Download MUTUAL ELECTROSTATIC INTERACTIONS BETWEEN

MUTUAL ELECTROSTATIC INTERACTIONS BETWEEN CLOSELY SPACED
CHARGED SOLDER DROPLETS
By Qingbin Liu, Changzheng Huang and Melissa Orme
Department of Mechanical and Aerospace Engineering
University of California, Irvine, CA 92697-3975
ACCEPTED FOR PUBLICATION IN JOURNAL OF ATOMIZATION AND SPRAYS
ABSTRACT
Emerging technologies of net-form manufacturing and electronic packaging rely on the use of
precisely deposited molten metal droplets with sizes on the order of 100 µm in diameter. In
many technological realizations, closely spaced droplets are electrostatically charged and
deflected onto a substrate in a manner similar to ink-jet printing in order to “print” fine features
onto a board for electronics applications or onto a substrate for net-form manufacturing. Unlike
ink-jet printing, the aforementioned emerging technologies require the printing of large lateral
dimensions onto the substrate by means of electrostatic charging and deflection (on the order of
several centimeters), and hence these applications require the droplets to have significantly
higher charges than in the ink-jet printing technology.
The high charges of the closely spaced
droplets can lead to inter-droplet electrostatic interactions that will cause significant deviations in
the droplets’ trajectories. Hence, the understanding of the physics of inter-droplet electrostatic
interactions is of primary importance in order to assure the fidelity of the net-formed component
or the printed electronic package. In this work, we present a model that predicts the trajectories
of the droplets when charged and deflected and corresponding experimental validations.
Conditions for which electrostatic interactions are negligible are sought.
The authors acknowledge the generous grant from the National Science Foundation Grant #DMI-9622400
which supported this work.
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NOMENCLATURE
A
a
b
Cd
D
E
Fi,i+1
f
g
I
i
k o∗
k o∗ max
m
cross sectional area of the droplet
distance between two deflection plates
radius of charge tube
drag coefficient
aerodynamic drag force
electric field strength
electrostatic repulsive force between droplets i and i+1
frequency of forcing disturbance
gravitational constant
current drawn by droplet stream
droplet index
nondimensional wavenumber =2πro /λ
nondimensional wavenumber at maximum growth rate
Q
Re
r
rd
ro
Vo
Vx
Vy
Vz
Vc
droplet mass
mass flow rate of droplet stream
droplet charge
Reynolds number
instantaneous distance between two droplets
droplet radius
initial stream radius
initial stream speed
component of velocity in x direction
component of velocity in y direction
component of velocity in z direction
charge potential
α
β
εo
λ
ν
ρ
ρa
σ
direction of repulsive force
growth rate of the applied disturbance
permittivity of free space
wavelength of the applied disturbance
kinematic viscosity of the fluid
density of molten metal
density of ambient fluid
surface tension of molten metal
m
&
2
1. INTRODUCTION
There is recent interest surrounding the use of highly controlled molten metal droplet streams for
net-form manufacturing of structural components [1-9] and printing of electronic packages [1012].
In a subset of these developing technologies, the droplet streams are electrostatically
charged and deflected for rapid high precision printing onto a substrate [1,3,7, 9-12]. Figure 1 is
a conceptual schematic of the droplet printing technology, which is implemented in this work.
Droplets that are generated from capillary stream breakup acquire an electrostatic charge by
passing through the charge tube at the point of droplet formation.
The initial direction of the
droplet stream is along the negative z-axis. The charge signal is carefully synchronized with the
droplet formation disturbance waveform so that droplets can be charged on a drop-to-drop basis.
The charged droplets pass through an electric field, which is established by two vertical
deflection electrodes as indicated in the figure. Passing through the electric field causes the
droplets to acquire a component of velocity in the x direction, the magnitude of which is directly
related to the charge on the drop. Continuous substrate motion in the y direction allows printing
in two dimensions as shown for the example of Ball Grid Array (BGA) fabrication for
electronics packaging. More details of the specific conditions employed to print the shown BGA
are provided in reference [12].
For the similar application of net-form manufacturing which
entails the fabrication of three-dimensional cohesive structural components without any postworking, droplets are deposited in an overlapping regime such that fluid and thermal interactions
occur, and successive layers are deposited in order to build the component in the z dimension.
Since many applications require droplet placement over large lateral areas, it is of interest to
determine the maximum charge that can be applied to the droplets before the onset of significant
inter-droplet interactions that prevent accurate droplet targeting.
Inter-droplet interactions occur
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since the droplets are closely spaced and carry like charges. Hence, the droplets act to repel each
other but their motion is impeded by the existence of their droplet neighbors on the opposite side,
which also possess charges of the same sign but different magnitude. Figure 2 illustrates the
phenomenon of mutual electrostatic interactions.
Shown are photographs of two molten solder
droplet streams issuing vertically into an inert environment from a 100 µm diameter orifice. In
each experimental realization the droplet generation frequency is 12 kHz and the initial vertical
droplet speed is 5.5 m/s. In the droplet trace shown to the left, the droplets were charged by
passing the capillary stream through a charge electrode at the time of droplet formation. A
constant voltage of 250 Volts DC is applied to the charge electrode, giving a charge of
-20.24×10-13 Coulombs to every droplet. The negative charge on the droplets is a result of using
a positive charge voltage. This causes electrons to be drawn onto the stream by the attraction to
the oppositely charged electrode. Since the droplets each posses the same charge, they act to
repel each other, however their motion along the vertical axis is constrained by the existence of
their charged neighbors on the opposite side. Therefore, the droplets move laterally out of the
main stream creating a helix spiraling down in x-y-and z, that appears to look like the zigzag
when viewed in one plane as shown. The current experimental apparatus is only equipped with
observation ports for photographing the x-z plane. Our interpretation of the helical behavior of
the charged droplet stream is based on our observations of the droplet trajectory after rotating the
droplet generator 90o in order to observe y-z plane. These observations also revealed that the
zigzag path was not limited to the x-z plane, but also occurred in the y-z plane.
While the
experimental results of this paper are limited to observations in the x-z plane, the phenomenon is
three-dimensional, and therefore a three-dimensional model is provided here.
The droplet
stream shown in the plate to the right is included as a reference of an uncharged solder droplet
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stream and was generated with the same conditions (excluding charge) as the droplet stream
shown to the left.
Since the charging waveform is unique for each pattern to be printed and is likely to be more
complicated than the DC example shown above, general methods of compensating for the
interactions are non-existent at this time and are not within the purview of this work. This work
is aimed at presenting a model which is corroborated by new experimental results that predicts
the droplet trajectory subject to highly charged, closely spaced droplets.
2. BACKGROUND
2.1.
Droplet Formation from Capillary Stream Break-up
Capillary stream break-up has been used to generate streams of uniformly sized droplets for well
over 100 years, and is typically achieved by applying the well-known Rayleigh instability [13].
In this work, the periodic motion of a piezoelectric crystal is used to establish a disturbance on
the surface of a molten metal capillary stream in order to initiate droplet formation as illustrated
in Figure 3. According to the linear theory of Rayleigh, the radial disturbance on the surface of
the jet will be unstable and grow resulting in droplet formation for conditions when the
nondimensional wavenumber, k o * , which is defined as the ratio of the initial stream
circumference to the wavelength of the imposed disturbance, is less than unity.
Lord Rayleigh developed the first linear stability analysis where he considered an infinitely long,
circular, inviscid jet subject to a temporal disturbance growth.
He found that disturbances of the
radius grow in time t as eβt, where β is the growth rate of the disturbance. A more detailed
description of the droplet formation process from capillary stream break-up can be found in the
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review articles by Bogy [14] and McCarthy and Molloy [15]. Droplet formation from capillary
streams has been the subject of numerous experimental and theoretical works.
A complete
review of the subject is not within the purview of this paper, which is focused on droplet charge
interactions.
The droplet generation technique defined by the Rayleigh instability is termed “continuous”
droplet generation since a capillary stream is broken into a continuous stream of droplets. Sheild
et al. [16] and Bousfield et al. [17] have developed techniques of generating droplets at given
pulses and thus are termed as drop-on-demand. The drop-on-demand mode of droplet generation
is limited by lower droplet production rates than the continuous mode.
2.2.
Droplet Charging
A great deal of background work on droplet charging has been performed in the 1960’s and 70’s
for the well-known application of ink-jet printing.
A few of the more important experimental
works are given by Sweet [18, 19], Schneider et al. [20], Kamphoefner [21], and Fillmore et al.
[22].
Analogous to the applications described in this work,
an ink-jet jet printer produces
characters on paper by deflecting charged droplets on one axis while the print head moves along
the perpendicular axis.
The droplet charge and the strength of the electric field through which
the droplet moves determine the amount of deflection achieved by a droplet. The net charge on
the droplet is acquired at the time of droplet formation.
In contrast to the ink-jet droplets, the molten metal droplets in this work attain significantly
higher charges so that large lateral areas can be printed in order to fabricate a structural
component of meaningful dimensions.
The high charges cause significant inter-droplet mutual
electrostatic interactions to occur which are not evident in the ink-jet printing technology as was
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illustrated in Figure 2.
A conceptual schematic of the droplet charging and deflection
configuration is shown in Figure 3. As previously discussed, droplets are charged by passing the
capillary stream through a charge electrode at the point of droplet break-off from the stream.
The molten jet is grounded, and a positive potential, Vc, is applied to the charge electrode. The
amount of charge given to the droplet is determined by the level of voltage applied to the
electrode at the time of droplet formation.
It is assumed that the conductivity of the liquid is
high enough to allow a steady charge to quickly form on the surface of the jet prior to break-up,
and that the charge acquired by each droplet is the surface charge on a length λ of the capillary
stream. With these assumptions, the charge to mass ratio can be predicted by the relation given
by Schneider et al. [20].
Q
2πε V
= 2 o c
m ρro ln( b / ro )
(1)
Here, m is the droplet mass, εo is the permittivity of free space, Vc is the charge potential, b is the
radius of the charge tube, ρ is the molten metal density, and ro is the radius of the unperturbed
capillary stream.
In order to attain 100% charge efficiency, the fluid attached to the capillary stream that will form
the leading droplet must reside in the charge tube for a time sufficient for the charge tube to
acquire the desired charge value.
After droplet formation, the voltage is changed in order to
apply a different charge on the next droplet to be formed. Since the conductivity of the liquid is
high, it is assumed that charge relaxation occurs faster than the transit time for an element of
fluid to move through the charge electrode. Since the droplets acquire a charge at the time the
droplet pinches from the capillary stream, the charge electrode should be positioned so that any
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excursions in droplet formation length are still contained in the charge tube. The synchronization
of the charge pulse with break-up time is adjusted manually by tuning the phase of the voltage
pulse to be synchronous with the break-up time. Excursions in break-up distance can occur due
to fluctuations in the driving pressure, fluctuations in amplitude of the disturbance due to the
resonance of unwanted frequency components that are apparatus specific, and variations in fluid
properties with temperature such as viscosity and surface tension. It is of critical importance to
eliminate any excursions in break-up distance since this will lead to a loss of synchronization
between droplet formation and droplet charge.
Hence, in the undesirable occurrence of
fluctuations in break-up distance, droplet charges will be “assigned” to the wrong droplets,
causing gross errors in droplet targeting.
3. MODEL DEVELOPMENT
Consider a sequence of similarly charged droplets emanating from a capillary stream that is
oriented along the z-axis.
Suppose that at time t=t1, droplet i experiences a disturbance that
causes it to drift a modest distance from the main stream. Then the nearest neighbors on both
sides of the drop, i+1 and i-1, immediately feel the repulsion forces from the droplet i, which in
return feels the anti-reaction force from its neighbors. Similarly, at a later time t=t2 , the droplets
i+1 and i-1 have been pushed out to the opposite side of the main stream. Then the droplets i+2
and i-2 feel the repulsion and they will move to the same side of droplet i. The process continues
until only a very few (if any) droplets will remain on their intended trajectory, deeming a
sequence of similarly charged droplets fragile to transverse disturbances.
The transverse
disturbance could be as simple as the droplet passing through a set of deflection electrodes for
the purpose of droplet targeting.
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A similar situation occurs when a stream of droplets carry different charges of high magnitude
(e.g., to print a horizontal line or other feature). A droplet stream in which each droplet has a
charge that is different in magnitude from its nearest neighbor, will also induce repulsive
reactions thereby causing them to adjust their positions.
The adjustment of one droplet position
is not an isolated event, since all of the droplets will adjust their positions to account for the
adjustment of their neighbors’ position. Hence, in the situation of non-uniform droplet charges
(or AC charging), the droplet excursions will not be the uniform helical behavior as shown in the
two-dimensional cross-section of Figure 2 for charges of equal magnitude and sign, but will be a
complicated sequence of droplet positions wholly dictated by the characteristics of the charging
waveform.
Governing Equations
The equations of motion used to simulate the trajectory of the ith charged droplet traveling
through an electric field of intensity E in the x direction are given as:
m
jmax
dVx,i
= − D cosθ x + ∑ Fij cosαx + Qi E
dt
j =1
(2)
j ≠i
m
m
dV y,i
dt
= − D cosθ y +
dVz ,i
= − D cosθ z +
dt
jmax
∑ Fij cos αy
(3)
j =1
j ≠i
jmax
∑ Fij cos αz + mg
(4)
j =1
j ≠i
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Where we assume that the undeflected droplet stream is aligned with the z coordinate which is
parallel to the gravity vector, and the deflection plates establish a field in the x-z plane so that the
magnitude of the deflection due to passage through the deflection plates is measured in the
perpendicular x coordinate. Ideally, droplet trajectories will deviate from the z-axis along the xaxis due to the established electric field.
However, mutual electrostatic repulsion will cause the
droplets to deviate from their ideal trajectory in both the x and y plane.
In the above, Vx, Vy and Vz are the components of velocity in the x, y, and z coordinates defined by:
cosθ x =
Vx
V
(5)
cosθ y =
Vz
V
(6)
cosθ z =
Vz
V
(7)
V 2 = Vx2 + Vy2 + Vz2
(8)
and the electrostatic repulsive force between the ith and j th droplet is given by:
Fij =
1 Qi Q j
4πεo r 2
(9)
Where Qi and Qj are the charges of the ith and j th droplets given by Equation (1). The angle α is
the direction of the repulsive motion given by:
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cosαx =
cosαy =
cosαz =
xi − x j
( xi − x j ) + ( yi − y j ) + ( zi − z j )
2
2
(10)
2
yi − y j
( xi − x j ) + ( y i − y j ) + ( zi − z j )
2
2
(11)
2
zi − z j
( xi − x j ) + ( yi − y j ) + ( zi − z j )
2
2
(12)
2
Where x i, x j, yi, yj, zi, and zj are the x, y and z positions of the ith and j th droplets with the z-axis
pointing vertically downward and the origin of the x and y-axes are placed at the top of the
deflection plates.
In the above, g is the gravitational acceleration, and D is the aerodynamic drag force defined by:
1
D = Cd ⋅ ρaV 2 ⋅ A
2
(13)
where ρa is the density of the atmosphere, A is the cross sectional area of the droplet, and Cd is
the drag coefficient of a sphere given by White [23]
Cd =
24
6
+
+ 0.4
Re 1 + Re
and Re is the Reynolds number based on the diameter of the droplet.
(14)
The drag coefficient
described by Eq (14) is for an isolated sphere traveling in a quiescent environment. Since the
droplets in this work are deflected by various amounts out of the main stream, the above model
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may overestimate the drag force since many of the droplets are protected by the wake provided
by their nearest neighbor.
The system of ordinary differential equations described by Equations (2), (3) and (4) above is
solved iteratively for each successive time step.
4. EXPERIMENTAL APPARATUS
The experimental setup consists of a droplet generator that injects droplets into an environmental
chamber, capable of being evacuated to a few times 10-5 torr with the aid of a diffusion pump.
Although the experiments are performed in an inert environment at a pressure of one
atmosphere, we have found that it is necessary to evacuate traces of oxygen from the chamber in
order to avoid the disruptive effects of oxidation which work to impede molten metal droplet
formation.
Immediately below the droplet generator are the charging electrode and the parallel
deflection plates. The separation between the orifice, which is contained at the lower end of the
droplet generator, and the substrate, is 295 mm. The substrate is mounted on an x-y table that is
capable of moving at speeds up to 13 cm/s.
allow optical observation in the x-z plane.
Two windows on opposite sides of the chamber
A schematic of the facility is shown in Fig 4. A
microscope with long working distance optics and equipped with a camera is mounted outside
the chamber. Values used in the experiment are given in Table 1 below unless otherwise noted
in the text.
The droplet deflection is highly sensitive to the initial unperturbed stream radius ro .
The initial
stream radius may not necessarily be identical to the radius of the orifice depending on the
orifice shape.
An orifice which has a smoothly contoured entrance region followed by a short
tube will have a relatively flat velocity profile at the orifice exit resulting in a stream radius
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droplet fluid
specific gravity
surface tension
Viscosity
orifice diameter
ambient and stagnation gas
driving pressure
solder reservoir temperature
disturbance frequency
charge electrode inner diameter
deflection plate separation
deflection plate length
Eutectic Solder, 63% Sn, 37% Pb
8.420
0.49 kg/s2
1.58x10-7 m2 /s
100 microns
Nitrogen
138 kPa
200o C
12,000 Hz
0.318 cm
1.27 cm
5.08 cm
Table 1: Properties used in experiment
commensurate with the orifice radius. Streamlines through a flat plate orifice, on the other hand,
will be curved at the exit, causing a non-uniform pressure distribution across the radius of the
stream.
The velocity distribution in the jet becomes uniform a short distance away from the
orifice where the jet area is smaller than the orifice area.
Stream radius measurements were made by measuring the overall mass flow rate of the droplet
stream, m& and relating it to measured stream quantities.
In each experimental realization, the
number of droplets collected was large enough to insure negligible errors due to turn-on/off
transients.
The disturbance wavelength λ, was measured by averaging up to 20 inter-droplet
spacings off of several video images. Knowledge of the driving frequency, f, and wavelength λ,
enables the estimate of stream speed V= fλ. From conservation of mass, the undisturbed stream
radius, ro , and the droplet radius, rd , are given by:
ro =
&
m
,
ρVπ
& 
3 m
rd = 

 4 ρπf 
(15)
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5. RESULTS
5.1.
Droplet Charge Measurements
As a benchmark to our deflection experiments, we have measured the droplet charge as a
function of charge electrode voltage. In order to measure the droplet charge the droplets are
collected in a conductive container that is set on an electrically insulated surface.
The container
is connected to ground through a picoammeter and the current is measured. The picoammeter is
connected between the droplet stream collector and the ground instead of between the voltage
source and charge electrode to avoid erroneously high current readings due to the leakage of
current through the thin insulator between the charge electrode and the grounded droplet
generator. The current measured corresponds to that drawn by the stream due to charging. The
following relationship is used to determine the droplet charge to mass ratio from our
measurements:
Q I
=
m m&
(16)
In the above, I is the current drawn by the droplet stream, and m
& is the mass flow rate of the
solder through the orifice.
Operating conditions for this measurement are as listed in Table 1
except for the fact that the charge electrode voltage is varied from zero to 200 volts DC. At the
largest charge potential used for this measurement of 200 volts, a current of -16.3 pA is drawn by
the droplet stream. This corresponds to an average droplet charge of -1.36×10-12 Coulombs. Our
experimental results are shown as circular symbols in Figure 5, which also shows the theoretical
prediction due to Schneider (Eq. 1) as a solid line. Note that agreement between experiment and
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theory is excellent, and we can therefore be assured that any deviations between predicted and
experimental deflection do not arise due to errors in charge efficiency.
5.2.
Charged Droplet Interactions
Figures 6 and 7 provide illustrative examples of the detrimental effects of mutual droplet charge
interactions.
In the figures, the characters “UCI” were printed by electrostatically deflecting the
charged droplets along the x-axis as shown while the substrate was in continuous motion along
the y-axis. The fluid dynamic properties of the droplets are provided in Table 1.
waveforms used to print the characters are provided in Figure 8.
The charging
The maximum charge used to
generate the characters in Figure 6 is -6×10-13 Coulombs, which is relatively low, and hence, this
charging waveform does not create observable errors in droplet targeting. The character height
is 4.7 mm. The characters shown in Figure 7, on the other hand, were obtained with relatively
high charges as is evident from the height of the characters (illustrating the extent of the
deflection) which is 9.4 mm. The maximum charge used to print the characters in Figure 7 is
–1.5×10-12 Coulombs. Also apparent in the figure is the undesired waviness of the lines formed
by droplet splats that create the characters.
This waviness is due to electrostatic interactions,
which are more severe when the droplets are highly charged as evidenced in the example shown
here. A few comments are required at this point to describe the nature of the deviations from the
ideal target for the high charging case.
First, it can be seen that there is not a systematic
displacement in droplet position with applied voltage that one may expect from the theoretical
expressions.
To understand this finding, we consider the waveform that was used to print the
character “U” as shown in Figure 8. The first five peaks illustrate the relative charges applied to
the five droplets that created the first of the two vertical lines along the x-axis of the “U”. One
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may expect the interactions between drops i=4 and i=5 to be greater than the interactions
between drops i=2 and i=3 due to the magnitude of their charges. It should be pointed out,
however, that drop i=5, which is a highly charged end drop, feels the repulsive force from only
one neighboring drop, i=4, whereas drops i=2 and i=3 feel the repulsive forces from
neighboring drops on both sides.
Hence, there is a competition between the magnitude of the
droplet charge and the magnitude of the charges of the neighboring drops.
Also, it appears that
the y-axis error is greater than the x-axis error. This is not necessarily the case, and the x-axis
error and the y-axis error are on a par.
The waviness (y-axis error) shown on the vertical
segments made from the five closely spaced charged drops is due to the helical behavior of the
droplet charges as they travel along z. However, there is also an error in the placement along the
x-axis, which is the y-axis error that is on a par in magnitude with the x-axis error that can be
observed by noting the relative differences in inter-droplet spacing along the x-axis.
Additionally, the base of the “U” character was printed with the two isolated droplets as shown
in Figure 8.
These droplets are sufficiently isolated to exhibit negligible inter-droplet
interactions, and the apparent error in droplet placement (which appears to be an x-axis error) is
somewhat misleading.
To illustrate this point, consider all of the droplets in the “UCI” except
those that were printed with closely spaced drops (i.e., except the four lines of five drops along
the x-axis).
Examination of the remaining drops illustrates that they exhibit very little x-axis
errors, and these positions are very near to their “ideal” positions. The gray horizontal line has
been included as a reference to illustrate the point that the only drops exhibiting errors from this
reference are those associated with the vertical lines of five drops. Hence, it is the lines that were
printed with closely spaced charged drops that exhibit mutual repulsions that cause the end drops
to “kick-out” along the x-axis farther from the “ideal” position.
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The model described in this work was developed as a tool to study the effect of electrostatic
mutual repulsion, and was used to simulate the trajectory of five droplets under the influence of
gravity, aerodynamic drag, and mutual electrostatic repulsive forces.
plane were also conducted to verify our model results.
Experiments in the x-z
Figure 9 illustrates the charging
waveform that was used in both the simulation and experiment.
In this case, the maximum
charge is –8.2×10-13 Coulombs which is considered to be relatively low. Figure 10 illustrates the
model simulation of the trajectories of five droplets subject to the charging waveform shown in
Figure 9.
excluded.
Here, the effects of gravity and drag are included, while the repulsive forces are
The circular symbols are our experimental measurements.
In both experiment and
simulation, the droplets traveled a downward vertical distance of 295 mm until impacting with
the substrate which is indicted by the horizontal dashed line.
In the absence of repulsive
interactions, the simulations predict that the droplets should be evenly spaced as shown where
the solid lines intersect the horizontal dashed line.
However, our experimental measurements
indicate that there is a small amount of repulsive interactions occurring as can be seen by the
increased separation in experimental data between drops i=3 and i=4. This is due to the fact that
the charges on the drops i=1 and i=2 are too small to cause measurable inter-droplet repulsions,
and the charges on drops i=3 and i=4 are larger than on any other bounded set of droplets
(droplet i=5 is unbounded and hence only feels the repulsive force from drop i=4). Including the
effects of mutual repulsion in the simulation leads to the results shown in Figure 11. In this case,
the model simulations are in excellent agreement with experiment, which are reproduced on
Figure 11 for comparative purposes.
Figure 12 illustrates the charging waveform used in the experiment and simulations for the high
charging case shown in Figures 13 and 14.
In Figure 13, the simulations included the effect of
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gravity and drag, but not electrostatic repulsion.
As can be seen by the solid lines, which depict
the trajectories of the droplets, the simulation predicts that the droplets will be uniformly
separated when they encounter with the substrate. Our experimental measurements show that the
droplet positions are highly non-uniform which is a consequence of the electrostatic repulsive
forces. Unlike the data of the low charging case as shown in Figures 10 and 11, the magnitude
of the charge on all of the droplets (even drops i=1 and i=2) is large enough to cause mutual
interactions, and therefore, the deviation between experimental and the ideal trajectories is
significant.
It can be seen that droplet i=3, the central droplet is the only droplet that hits its
target. This is because, unlike the case for low charging, the interactions from both drops i=2
and i=4 are significant, and the deviations become more symmetric (though not completely
symmetric due to the variations in absolute droplet charge).
The discrepancy between
experiment and simulation here illustrates the magnitude of the repulsive forces.
Figure 14
illustrates the simulated droplet trajectories with the effect of gravity, drag, and electrostatic
repulsion included.
As is evident from the figure, there is excellent agreement between
experiment and simulation, and electrostatic interactions cause significant deviations from the
ideal droplet trajectories.
It is worth noting that without considering the effects of repulsive
forces in the simulation, one may be led to believe that there is scatter due to experimental error
in the measurement. However, the consideration of the effects of the repulsive forces allows the
correct conclusion to be drawn that there is indeed little experimental error and the droplet
placement can be predicted to a high degree of accuracy.
There are two distinct issues that define upper limit for the charge that can be applied to a drop.
The first, which is the subject of this paper, defines the maximum charge that can be applied to
the stream before mutual charge interactions cause unacceptable printing resolution.
In this
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work, we assume that droplet placement deviations greater than ±12.5 µm at the substrate
resulting from charge interactions are unacceptable as suggested by individuals representing
certain sectors of the electronics assembly industry [12].
The second issue is that of charge
induced droplet disintegration. If the electrostatic forces are strong enough to overcome the
surface tension force that holds the drop together, the drop will catastrophically disintegrate.
Lord Rayleigh [24] estimated the maximum charge before disintegration with the following
expression:
Qmax = 64π 2 ε o rd3 σ
(17)
For the molten solder droplets generated in this work, Qmax is equal to 48.0 ×10-12 Coulombs,
which is well above the values used in this work.
We have used the model presented in this work to estimate the maximum charge that can be
applied to a droplet stream before mutual electrostatic interactions cause deviations in placement
on the substrate greater than ±12.5 µm. The deviations in trajectories depend on flight distance
from the exit of the deflection plates and the magnitude of the deflection field. In order to make
the analysis independent of our apparatus dimensions, we have defined the origin of the
coordinate system to be coincident with the exit face of the deflection electrodes, and plotted the
maximum charge as a function of flight distance for several values of the deflection plate
voltage. The results are shown in Figure 15, where it can be seen that the mechanism governing
the limiting droplet charge is due to mutual droplet interactions and not to droplet disruption, as
the maximum charge for droplet disruption is over an order of magnitude higher than that for
unacceptable interactions.
Since the targeting errors due to interactions amplify over flight
distance, the maximum charge that may be applied to the drop decreases with flight distance. It is
19
also seen that droplets traveling through a weaker electric field can tolerate higher charges before
significant interactions are apparent. It is shown in previous work [25], that the droplet deflection
is a function of the product of the charge voltage and the deflection voltage. Hence it can be
reasoned that the errors in droplet deflection also depend on the product of droplet charge and
charge potential, and therefore, an increase in droplet charge can be compensated for by
decreasing the deflection potential as evidenced by the simulation results in Figure 15.
For
conditions employed in this work, the simulations indicate that the charge must not exceed
–6.1×10-13 Coulombs for a deflection voltage of ±2,500V if droplet placement accuracy less than
±12.5µm is to be maintained, which is consistent with all of the results presented in this work.
The model provided in section 3 could be used to predict the maximum charge for any required
printing accuracy.
6. CONCLUSIONS
Through model simulations and experimental measurements, it is shown that electrostatic
repulsive forces between highly charged droplets are significant and can cause considerable
deviations from the ideal droplet trajectory, thereby leading to printing errors if the repulsive
forces are ignored.
Such errors are not apparent in the technology of ink-jet printing since
smaller deflections are required, deeming the use of high charges unnecessary. Hence, it is the
emerging applications of net-form manufacturing and electronic package fabrication from metal
droplet printing, which rely on the deposition of highly charged metallic droplets
in order to
print large dimensions onto a substrate that will suffer from the existence of electrostatic
repulsive forces. In those applications, methods to compensate for the repulsive forces must be
developed.
Since the series of repulsive interactions for a droplet stream is unique for each
20
charging waveform, optimization of the charge waveform in an effort to compensate for the
interactions can be a tedious endeavor since general optimization methods are currently not
available.
Also presented was a prediction of the maximum charge that can be applied to the stream before
electrostatic interactions that cause deviations greater than ±12.5µm in droplet placement at the
substrate occur, which is a requirement imposed by certain sectors of the electronic package
industry. The model presented in this work can also be used to ascertain the maximum charge
for any required printing accuracy.
It is shown that the maximum charge that can be applied
decreases with flight distance to the substrate, which can be reasoned to be due to the fact that
the errors in droplet trajectory are amplified with flight distance.
Also, it is shown that
increasing the magnitude of the electric field through which the charged drop travels results in a
decrease in the maximum charge that can be applied to the drop. For the conditions employed in
this work, the limiting charge is due to repulsive interactions, and the maximum charge before
droplet disintegration is nearly two orders of magnitude higher.
21
1. Orme M., Muntz E.P. 1992, United States patent Number 5,171,360, December 15
2. Sachs E., Cima M., Bredt, J., and Curodeau, A., “CAD-Casting: The Direct Fabrication of
Ceramic Shells and Cores by Three Dimensional Printing,” Manufacturing Review, Vol. 5, No.
2, pp 118-126, 1992
3. Orme M., 1993, “A Novel Technique of Rapid Solidification Net-Form Materials Synthesis,”
Journal of Materials Engineering and Performance, 2, (3)
4. Prinz, F.B., and Weiss, L.E., 1994, “Method and Apparatus for Fabrication of ThreeDimensional Metal Articles by Weld Deposition,” U.S. Patent No. 5,207,371
5. Chin, R.K., Beuth, J.L., and Amon, C.H., 1995, “Control of Residual Thermal Stresses in
Shape Deposition Manufacturing, Solid Freeform Fabrication Symposium, Austin, TX, 221-228
6. Argarwala M.K., Van Werren R, Jamalabad V., Langrana N., Whalen P., Danforth S.C. and
Ballard C. “Quality of Parts Processed by Fused Deposition” Proceedings to the Solid Freeform
Fabrication Symposium, University of Texas at Austin, 1995
7. Orme M, Huang C and Courter J, 1996, “Precision Droplet Based Manufacturing and
Material Synthesis: Fluid Dynamic and Thermal Control Issues”, ILASS Journal of Atomization
and Sprays vol. 6
8. Amon, C.H., Schmaltz, K.S., Merz, R., Prinz, F.B., 1996 “Numerical and Experimental
Investigation of Interface Bonding Via Substrate Remelting of an Impinging Molten Metal
Droplet” ASME J. of Heat Transfer, 118, 164-172
9. Orme, M.E., Huang, C. 1997, “Phase Change Manipulation for Droplet-Based Solid
Freeform Fabrication”, ASME Journal of Heat Transfer, 119, 818 – 823
10. Gao F. and Sonin, A.A., 1994 “Precise deposition of molten microdrops: the physics of
digital microfabrication,” Proc. R. Soc. London, Ser. A 444, 533
11. Prinz, F.B., Weiss, L.E., and Siewiorek, D.P., 1994 “Electronic Packages and Smart
Structures Formed by Thermal Spray Depositions,’ U.S. Patent No., 5,278,442
22
12. Muntz E.P., Orme, M.E., Pham-Van-Diep, G., and Godin, R. 1997 “An Analysis of
Precision, Fly-Through Solder Jet Printing for DCA Components’ Proceedings of the 30th
International Symposium on Microelectronics, Pennsylvania , October 1997
13. Rayleigh, Lord, 1879, On the instability of jets. Proc. London Math. Soc. 10, 4-13
14. Bogy, D.B., “Drop Formation in a Circular Liquid Jet” Ann. Rev. Fluid Mech., 1979, 11:
207-228
15. McCarthy and Molloy, “Review of Stability of Liquid Jets and the Influence of Nozzle
Design’ Chem. Engineering, 7, 1-20, 1974
16. Shield T.W., Bogy D.B., and Talke F.E. “Drop Formation by DOD Ink-jet Nozzles: A
Comparison of Experimental and Numerical Simulation,” IBM J. Res. Dev. 31, 96, 1987
17. Bousfield D.W. and Denn M.M. “Jet Breakup Enhanced by an Initial Pulse,” Chem. Eng.
Comm. 53, 61, 1987
18. Sweet R. G. “High-Frequency Oscillography with Electrostatically Deflected Ink Jets,”
Stanford Electronics Laboratories Technical Report No. 1722-1, Stanford University, CA, 1964
19. Sweet R. G. “High Frequency Recording with Electrostatically Deflected Ink Jets”, Rev.
Sci. Instrum. 36, 2, 131, 1965
20. Schneider J.M., N.R.Lindblad, and Hendricks C.D, “Stability of an Electrified Liquid Jet”,
J Applied Physics. 38, 6, 2599, 1967
21. Kamphoefner F.J. “Ink Jet Printing”, IEEE Trans. Electron Devices ED-19, 584, 1972
22. Fillmore G.L., Buehner, W.L., West, D.L., “Drop Charging and Deflection in an
Electrostatic Ink Jet Printer”, IBM J. Res. Develop. Jan, 1977
23. White F. M. Viscous Fluid Flow, 2nd Edition, McGraw-Hill Inc. 1991
24. Lord Rayleigh, Phil Mag. 14, 184 (1882)
25. M. Orme, Q. Liu, J. Courter, and R. Smith, “Electrostatic Charging and Deflection of
Arbitrary Molten Solder Droplet Formations” Physics of Fluids, A, (in review) 1998.
23
FIGURES
Figure 1: Schematic of droplet printing with electrostatically charged and deflected molten
metal droplets.
The Ball Grid Array shown was printed in 0.14 seconds from a
stream of droplets generated from capillary stream breakup.
Ball diameters are
approximately 190 microns.
Figure 2: Photograph of two molten solder droplet streams. The top droplet in each plate has
traveled 90 mm from the orifice exit. The droplet stream on the left is composed of
droplets which charges of 20.24×10-13 C, and the droplets on the right are uncharged.
Figure 3: Photograph of molten solder droplet formation from capillary stream break-up and
illustrative sketch of generation and charging phenomenon.
Figure 4: Schematic of experimental apparatus
Figure 5: Measured charge on a solder droplet stream as a function of charge electrode voltage
compared to the theory of Schneider.
Figure 6: Molten solder splats composing the characters “UCI”.
Droplets that were 189
microns in diameter were charged and deflected onto a black paper substrate where
they underwent the splatting action as shown. Droplets were charged with fairly low
charges as is evident from minimal charging interactions.
Figure 7: Molten solder splats composing the characters “UCI”.
Droplets that were 189
microns in diameter were charged and deflected onto a black paper substrate where
they underwent the splatting action as shown.
Droplets were highly charged as is
24
evident from the size of the characters and the charging interactions, which cause
observable non-uniform character spacing.
Figure 8: Charging waveforms used to print the characters U, C, and I as shown.
The
maximum charge used in the low and high charging (figures 6 and 7 above
respectively) case is –6×10-13 and –1.5×10-12 coulombs respectively.
Figure 9: Charge waveform used in the experiments and simulations shown in Figures 10 and
11.
Figure 10: Model simulations shown as solid lines of the trajectories of five droplets for low
charging with gravity and drag included.
Circular symbols illustrate experimental
results.
Figure 11: Model simulations shown as solid lines of the trajectories of five droplets for low
charging with gravity, drag and electrostatic repulsion included.
Circular symbols
illustrate experimental results.
Figure 12: Charge waveform used to generate the experimental results shown in Figures 13 and
14.
Figure 13: Model simulations shown as solid lines of the trajectories of five droplets for high
charging with gravity and drag included.
Circular symbols illustrate experimental
results.
25
Figure 14: Model simulations shown as solid lines of the trajectories of five droplets for high
charging with gravity, drag and electrostatic repulsion included.
Circular symbols
illustrate experimental results.
Figure 15: Maximum charge that can be applied to the solder droplet stream before interactions
cause deviations greater in value than ±12.5µm at the substrate. The three curves
correspond to values of the deflection plate voltage of ± 1500, 2500, and 3500 volts
with a 1.27cm separation between deflection plates.
26
Inert
environment
Molten
droplet
stream
Charge
electrode
Deflection
plates
Substrate motion
27
28
orifice plate
charge tube
vibrating rod
2rd
b
λ
λ
2ro
λ
droplet formation signal
charge s ignal
29
30
31
x, deflection axis
y, substrate
motion
32
x, deflection axis
y, substrate
motion
33
34
35
36
37
38
39
40
41