Electrohydrodynamic (EHD) excitation of liquid jets offers an alternative to piezoelectric excitation without the complex frequency response caused by piezoelectric and mechanical resonances. In an EHD exciter, an electrode near the nozzle applies an alternating Coulomb force to the jet surface, generating a disturbance which grows until a drop breaks off downstream. This interaction is modelled quite well by a linear, long wave model of the jet together with a cylindrical electric field. The breakup length, measured on a 33 micrometer jet, agrees quite well with that predicted by the theory, and increases with the square of the applied voltage, as expected. In addition, the frequency response is very smooth, with pronounced nulls occurring only at frequencies related to the time which the jet spends inside the exciter

Crowley, J. M.

English

NASA Technical Reports Server

Electrohydrodynamic (EHD) Jtimulation of Jet Breakup 
Joreph M. Crowley 
Dept. of Elec. Eng., Univ. of Illinois, Urbanat Illinoir 
and Xerox Corporation, Rocheater, New York 
Abrtrrct 
Electrohydrodyanmic (EHD) excitation of liquid jet8 offerr an alternative to 
piezoelectric excitation without the complex frequency reaponre caured by piezoelectric 
and mchanical reronances. In an EHD exciter, an electrode near the nozzle applier an 
alternating Coulomb force to the jet surface, generating a dirturbance which grow8 until a 
drop breaks off dcwnrtream. This interaction can be modelled quite well by a linear, long 
wave model of the jet together with a cylindrical electric field. The breakup length, 
mearured on a 33 pm jet, agree8 quite well with that predicted by the theory, and 
increases with the aquare of the applied voltage, a# expected. In addition, the frequency 
responre ir very smooth, with pronounced nulls occurring only at frequencier related to 
the time which the jet rpendr inride the exciter. 
Introductioq 
Mort of the existing ink jet printers bared on jet breakup uoe acoustic excitation of 
the jet to introduce the dirturbance which eventually grow8 to form ink drops. With a 
ringle jet, this method works very well, rince the breakup point can be controlled through 
a feedback loop which alters the amplitude of the excitation. The extenrion of this 
technique to multiple jet print9rs prerentr rome difficulties, however, tinct the jet8 
interact with the acou8t?~ waver rat up in the exciter, causing variations in the breakup 
length. 
Excitation baaed on electrostatic forcer appears to offer more promire for multiple jet 
prinkerr, rince the excitation ir applied via electroder which are placed downatrean of 
the nozzle, allowing individual control of individual jete. Although droplet production 
by electric excitation ham been known for some time, there haa been no work reported which 
offerr a theoretical model of thin procerr, or which dewribes r y r t e ~ t i c  warureaentr of 
the brerkup length with EHD excitation. The present paper ir intended to fill this gap. 
A Simple Theory oi E B  Excitation 
- 
A detailad analyrir of the EHD exciter ham been undertaken, and will be prerented 
e l e w e  . In the course of that work, a simple model war developed, which will be ured 
here to describe the exciter ured in the experirntr. The geometry of the jet in defined 
by Figure 1, which shows a liquid jet entering an electrode. The jet hrr a radiur R, and 
roves with a velocity no. 
Figure 1 
A Liquid jet entering an electrode 
Figure 2 
Electric prerrure in the rtep exciter 
https://ntrs.nasa.gov/search.jsp?R=19820015553 2020-03-21T16:53:13+00:00Z
z 
If. 
Thin particular electrode i r  rhown aa a cylinder, but it can have any rhapo, rince the 
theory arruner only that the electr ic  field a t  the rurface of the jet can k calculated. 
2 Theoretically, the jet i r  nodelled by the long wave apptoriut ion , in which the 
dirturbsnce is arrumed to bo much longer than the jet radiu8, and The axial velocity and 
prerrure are arrumed to be conrtant acro3r any section of the jet,  With there ' 
arrumptionr, the aronenttcc-, &;idation for motion i n  the axial directio,r can b. written a8 
while Conrervation of Mar8 for a round jet giver 
The pressure i n  the cDosclntum equation conri-ts of contribution8 from the rurface 
tenrion of the jet and a180 from the electr ic  field a t  i t .  rurface. 
The Inertial L i m i t  
--- -
The surface tension term i s  well known i n  the fluid wchanicr of jetr t  it leadr to 
growth of the dirturt,ance into drops. Inqide the exciter, which i l  usually only a 
fraction of a wavelengt',? long, the droplet growth is r - ry  small, 80 tlra effects of rurface 
tension w i l l  be negltkcted here. m.e electr ic  prerrure can be most earily obtained by 
u s i n g  the Maxwell s t rerr  tensor a t  the rurtace of the jet. Since the jet i r  arrumed t o  be 
a gaod e l ~ c t r i c a l  condustor, the electr ic  field w i l l  be ngrmal to the rurface of the jet,  
which implier that no electr ic  mhear forcer can be exerted . The only e l rc t r ica l  force 
corresponds to a (negative) pressure, given i n  terar of the field 8' rength a t  the jet 
surface as 
For example, if the electrode surrounding the jet i s  assumed to  bet a circular cylinder of 
radiur b, a t  a voltage V(x,t),  then the normal electr ic  field a t  the n:rrface of the jet 
w i l l  be 
so that the e l t c t r i c  pressure is given by 
Linearization 
------ 
In m r t  of the growth region the amplitude of the waver I8  usually quite rmall, uo that 
the equation8 may be further rimplified by linearization. I f  we define 
and neglect a11 second order terar,  the equatlonr of motion for r u r i  6,.3tutSances on the jet reduce to 
There equations have k e n  written in terms of the convective derivative of the surface 
displacement, defined by 
In the inertial approximation, the use of this surface velocity simplifies the resulting 
equations. 
m e  electric pressure depends on the electric field at the surface of the jet, and this 
field will usually change as the jet expands and contracts. Thus perturbattons in radius 
will lead to perturbations in the electric pressure, which in turn depend on the detailed 
structure of the field near the jet. Like the surface tension term, the perturbation 
electric field mainly influences the growth rate of the droplet disturbance. Its effect 
is usually much less than that of the surface tension, so it will also be neglected, as 
the sdrface tension was. The equation of the jet then becomes 
The linear response of the jet will generally be expressed in terms of its response to 
s sinusoidal pressure variation. For each term in the Fourier series, the driving 
pressure is most conveniently written i , ~  terms of compl~v amplitudes as 
The response to this drive can litewiss be written as 
so that the equation of motion for the exciter (Equation 12) reduces to an ordinary 
differential equation for the complex amplitude of the radial velocity 
There are several ways to solvc this equation, but since we will be dealing with 
singular spatial functions such as the impulse and step, the easiest solution is that 
which uses the Laplace transform in x. With the de-inition of the Laplace transform as 
and 
the equatian can he transformed into 
a 2 -  (iw + UOs) 6 '  = - 2p 
which can be inverted to find the response as soon as the spatial distribution of electric 
pressure is known. 
Constant Pressure Ebtciter 
The simplest practical exciter is a cylindrical electrode of length R . The entire 
exciter is driven from a single voltage source, for example V +V sin(w t) , so that the 
same pressure is applied throughout. For this exciter, p has th8 sbatial form shown in 
Figure 2. with a time varying magnitude of 
2 
p ( t )  = - - [iV: - $1 + 2vlv2 s i n  wt + 2 sin 2.t 
2a2 p n 2 h / a )  I 
This pressure has compon*nts at DC, o, and a t  Zw, but since the equation is linear, each 
component can be treated separately. For the component at w, the electric pressure has 
the form 
where V is the effective (RMS) value of the voltage, which in this case is 2VoV1. Por 
this excP€br, the Laplace transform of the drive is 
giving the radial velocity at the exit as 
The response has an impulse (A) in velocity beause the jet quickly contracts as it 
enters the exciter and expands as it leaves. These impulses have no effect downstream, 
since they vanish as soon as the jet has passed the entrance or exit. For practical 
purposes, then, the velocity of the jet at the exit of the exciter can be written as 
The velocity hae an exponential delay tern, which can be analyzed more conveniently by 
extracting a factor corresponding to the center of the exciter, using the relation 
iwe ~ W P  
- -  - -  
= e  
wL 1 - e  21 s i n  - 2 
The velocity response then has the magnitude 
rwp 
1 6 3  - 3 Isit. 
"0 
I, .  * 
--... 
is ' '  - 
4 
ol S W  WUE 
W W A L l f t D  LENGTH 
Figure 3 
Exciter output versus length 
Figure 4 
Force variation within the exciter 
which is shown in Fiqure 3 as a function of exciter length. 
The nulls and peaks which occur in the response are caused by the alternation of 
electric pressure while the jet is inside the exciter. If the pressure is sinusoidal, as 
shorn in Figure la, the net impulse given to the jet over one period of excitation will be 
zero, since the jet 1s influenced by equal and opposite half cycles. Thus if the jet is 
within the exciter for a whole period of excitation, (i.e., the exciter is one wavelength 
long), the response will show a null. On the other hand, there will be a peak for lengths 
which are one half wavelength long. While this effect is easy to see from Figure la, some 
care must be used in interpreting such diagcams, since they may suggest erroneous nulls, 
depending on the choice of phase at the inlet. For example, if the sinusoidal force were 
chosen to be a cosine, as in Figure 4b, it would be easy to infer that the null would 
occur when the exciter is only one half wavelength long. This 1s not true, because the 
coaine is one special cho-cr from a11 of the phases which occur during a conpl-ete cycle. 
Because of the peaks and nulls in the frequency response, the output of the exciter can 
not be increasefl indefinitely simply by increasing the length of the exciting electrodes. 
Instead, an optimum length, which may be fairly small by ma:roscopic standaras, must be 
selected to achieve maxiaum output. This explains why the electrodes used in earlier work 
on EFII, excitation were usually not very efficient. 
The approximate surface displacement far downstrerm is given by 1 
s 
where 
: The breakup length can be determined by extrapolating the downstream response to the point 
of breakup, tg to give the breakup length equation, 
Thus the breakup length can be calculated as soon as the exciter output is known. 
Rxpcr iacnts 
Apparatus 
Several experiments were performed in order to test the predictions of the BHD exciter 
theory. The nozzle and associated app3ratus were mounted on a specially designed frame 
which allowed adjustment and positioning suitable for tge experiments. The fluid used in 
moat of these experiments had a density of 1030 kg/m and a surface tension of 39 W/r. 
The jet formed by the nozzle had a radius of 16.5 um. Its velocity was determined by 
measuring the wavelength of the disturbance at a known frequency. 
The exciter electrodes used in most of these experiments consisted of steel plates of 
various nominal thicknesses (3-7 mil, or 75-175 urn) through which holes were drilled. The 
holes ranged in nominal diameter from 3-7 mils (75-175 urn). nicroscopic examination of 
the electrodes revealed that the holes were essentially straight sided, although 
occasional burrs could be seen. 
Since the e:ectrodes consisted of a single piece of metal, they had to be in position 
before the jet was turned on to avoid splashing and electrical shorting between exciter 
and ground. As a compromise between speed and complexity, we first located the 
approximate axis of the jet by focussing a microscope on the noxxle orifice along the 
nozzle axis. The electrode, mounted on a three axis nicroaanipulator, was then roved into 
position so that the desired exciter hole appeared to be on the nozzle axis, as seen 
through the micros cop^^. The microscope was then removed to the side, where the jet and 
hole would both be visible at an oblique angle, and the jet turned on. The original 
electrode placement was usually within a few mils of the desired placement, could be 
quickly adjusted to center the electrode and allow the jet to stream through unhindered. 
The jet was then turned off to allow the electode to be cleaned by wiping with 
absorbant paper towels. Upon restarting, the jet always went through the hole cleanly, 
without sprashing or blocking, even for the 3 mil hole, once the initial centering 
procedure had k e n  carried out. In most cases, the jet would restart cleanly even if the 
electrode had not been wiped clean. 
Several arrangements were used to provide different voltage levels. An operational 
amplifier could be connected directly to the electrodes, giving voltage up to 150V. In 
order to increase the voltage, the output of the operational amplifier was often connected 
to either a wide band or a narrow band transformer. The secondary winding of the 
transformer furnished higher AC voltages than the amplifier, and additional increases in 
the effective voltage could be achieved by connecting thr secondary in series with a M: 
power supply. With this arrangement, peak voltages up to 880V were obtained. This 
voltage was not limited by breakdown in the exciter, but by the available power supplies 
and amplifiers. 
In carrying out the'experiments, the jet was turned on and the voltages ad$usted to the 
desired value. The position of the breakup was then measured by a micrometer waich roved 
a long focal length microscope to the breakup point while the phaqe of the viewing strobe 
was slowly rolled. The breakup position can be measured very precisely (+lo um) with this 
technique if the jet is quiet. At very long breakup lengths ( > I  m), the natural breakup 
of the jet caused the breakup point t3 vary erratically. Lateral movement of the 
electrode varied the breakup point, probably by increasing the electric field strength at 
the surface of the jet. This effect was used to center the jet before each measurement, 
since the point of minimum excitation should correaponcl to a centered jet. 
Frequency Response 
M e  simple theory presented above predicts maxima and minima in the frequency response 
of the exciters. In order to test these predictions, frequency sweeps were performed on a 
number of exciters. The easiest feature of the response to check is the null which occurs 
when the frequency corresponds to an entire wavelength of the disturbance inside the jet 
at any time. Figure 5 depicts the frequency response for an electode which is 7 mils (178 
pm) thick. 
9 1  : I r - 1 - 1  
EXP NULL 
t 
0 20 40 60 80 I 0 0  120 140 I60 I80 200 
FREQUENCY ( k Hz 
Figure 5 
Frequency response for a 7 mil exciter 
EXP. NULL 
4 
0 1  . . . I 
0 20 40 60 60 I00 120 I40 160 180700 
FREQUENCY ! kHz) 
Figure 6 
Same thickness, but 8 mil theory 
For this length, the null is expected at 108 kHz. Since this nail occurs near the maximum 
growth rate of the jet, it could be ohserved experimentally by varying the frequency to 
produce the longest apparent hreakup length. This null frequency (shown in Figure 5.3 by 
an arrow) is lower than expected from theory by approximately 15%. Reasonable variations 
in the fringing length and jet velocity could not accolnt for the discrepancy, but 
increasing the effective length of the exciter from 7 to 6 mils gave better agreement, as 
shown in Figure 6. This extension in length might be justified by the presence of burrs 
at either end of the hole, since such burrs were often observed in microscopic examination 
of the holes. bnother possibility is the effect of fringing in extending the effective 
length of the electrode beyond the physical limits of the conductor. 
This exciLor exhibits the predicted null in the frequency response, but does not test 
the exciter at the thickness which is expected to jive the shortest breakup length. Since 
short breakup length is desired in practical printers, the frequency response of a shorter 
(3 mil or 76 )am) electrode was also measured. The results of this measurement, along the 
the predictions of the theories, are shown in Figure 7. 
J*t:~t 4s !n t h e  earlier measurements, the theory predicts both the magnitude and the 
shape of the frequency response quite well (t0.2 mm or 108) whether the fringing fields 
art? included or not, although the Cringing approximatton seems to give better results at 
the shortest breakup lengths. '5 appreciate this agreement, it should be kept in mind 
that the theory, which rests on the fundamental eq~ations of electrostatics and fluid 
mechanics, contains no adjustab?c parameters. The breakup length is predicted only in 
terms of geometrical measurem.*nts, applied voltage, and material properties. 
Voltage response 
Compared to acoustic excitation, EHD excitation is often relatively weak, so #:hat the 
magnitude of the drive is extremely important in practice. The theory presented above 
predicts that the excitation pressure is proportionti1 to the square of the effective 
voltage, so that a relatively large increase in the excitation may be! obtained by 
increasing the voltage at which the exciter operains. This prediction was tested 
experimentally by varing both the AC and DC voltage levels over a wide range, and 
measur inq the breakup length. 
30% 1 
50 100 M LOC 
FREQUENCY ( knt ) 
Figure 7 
Frequency response for a 3 mil thickness 
THEORY 7-22 a 
I f = l I0 k H z  \ I = 3 mil vo = I8 3 m/s 0 J 1 _ - 
10' 10' 1 oe 
vtff ( VOLTS * 
Figure 8 
Breakup length versus effective voltage 
Since the breakup length depends logarithmically on the excitation, a plot of breakup 
length against the logarithm of the effective voltage should give a straight line. 
Experimental values of 3reakup length are plotted in this way in Figure 8, along with the 
predictions of the simpe theoretical model. Both the magnitude and slope of the breakup 
length follow the predictions well, although the slope of the line appears to be somewhat 
flatter than expected. These results give us some confidence in extrapolating the design 
to even higher voltages to achieve a shorter breakup length if necessary, although these 
lengths are already comparable to those used in acoustic excitation. 
Acknowledgments 
The experinental work described here was carried out at the laboratories of the Xerox 
Corporation at Webster, Yew York, on occasional visits. Its success is due in large 
measure to the active cooperation of several people, especially Thomas Warren, in seeing 
that the necessary equipment was set up and working, and that help was always available 
when needed. 
Reference8 
1, CroWley, J. M., "The Theory of Electrohydrodynamic Droplet Generators," to be 
pub1 a thed 
2. Lee, 8. C., "Drop Formation in a Liquid Jet," IBI Journal of Research & 
Development, July, 1974, pp.364-369 
3. Melcher, J. R., Field Coupled Surface Waves, HIT Presa, Cambridge, Mass., 1963 


Electrohydrodynamic (EHD) stimulation of jet breakup

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