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APPLIED PHYSICS LETTERS 104, 024103 (2014)
The minimum flow rate scaling of Taylor cone-jets issued from a nozzle
William J. Scheideler and Chuan-Hua Chena)
Department of Mechanical Engineering and Materials Science, Duke University, Durham,
North Carolina 27708, USA
(Received 13 July 2013; accepted 30 December 2013; published online 15 January 2014)
A minimum flow rate is required to maintain steady Taylor cone-jets issued from an electrified
nozzle. This letter reports that the well-known scaling law proportional to the charge relaxation
time and independent of the nozzle diameter is only applicable to lowly viscous systems. For
highly viscous systems with rapid viscous diffusion, the cone-jet transitional region spans a length
scale comparable to the nozzle diameter; the minimum flow rate appears to be proportional to the
capillary-viscous velocity and the cross-sectional area of the nozzle, but independent of the liquid
C 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4862263]
conductivity. V
The electrohydrodynamic Taylor cone-jet1–4 is fundamental to a variety of applications including electrostatic
spraying, spinning, and printing. In the most common setup, a
working fluid is pumped at a flow rate Q through a round nozzle, which is electrified at a voltage V with respect to a counter electrode plate using air as the surrounding medium.
Steady cone-jets are known to exist only for a limited stability
window in the voltage-flow rate (V – Q) operating diagram,
where the applied voltage is around the threshold giving rise
to the Taylor cone2 and the imposed flow rate is above a minimum threshold necessary to sustain a steady jet.3 Although
the applied voltage (field) can be rationalized by balancing
the Maxwell and capillary stresses as in the case of the hydrostatic Taylor cone,2,5–7 the minimum flow rate required to sustain a steady jet is much less well-understood.4,8–13 In
electrospraying systems with the common setup of a working
fluid pumped through an electrified nozzle, the minimum flow
rate (Qm) is known to scale as4,14–16
Qm ec
;
rq
(1)
where e, c, r, and q are, respectively, the permittivity, surface
tension, electrical conductivity, and density of the working
fluid. This scaling law works well for the electrospraying of
liquids in the “high-conductivity” limit ðrⲏ1 lS=cmÞ,4 and
can be justified on dimensional grounds as long as the liquid
viscosity (l) drops out of the picture. Note that the length
scales of the overall system such as the nozzle diameter do
not enter the scaling law in Eq. (1), because the jet diameter
in the high-conductivity limit is typically orders of magnitude
smaller than the nozzle diameter for high-conductivity
liquids. The requirement of a minimum flow rate is believed
to give rise to pulsating cone-jets when the supplied flow rate
is smaller than this minimum value.17–20 In this sense, the
minimum flow rate scaling is relevant to the transient (quasisteady) cone-jets issued from solid nozzles18,19,21–27 as well
as liquid drops17,28–31 and films/pools.32–34
The absence of liquid viscosity in the minimum flow
rate scaling Eq. (1) lacks a rigorous justification, which poses
a severe limitation considering that highly viscous cone-jet
a)
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0003-6951/2014/104(2)/024103/4/$30.00
systems are critical for many practical situations. In nanoelectrospray, nozzles of micrometric or sub-micron diameter
are routinely used;23,35–37 In electrospinning, viscoelastic
polymeric fluids are typically employed;7,9,38,39 In electrohydrodynamic printing, fine nozzles are often used, sometimes
in combination with viscous working fluids.40–44 Despite the
practical relevance, systematic studies of the effects of viscosity on Newtonian cone-jets are scarce, and the few available studies have not focused on the minimum flow
rate.34,45,46 Here, we study the role of viscosity (l) in determining the minimum flow rate (Qm) for Taylor cone-jets
issued from an electrified nozzle. We show that the conventional Qm scaling Eq. (1) breaks down for cone-jets of highly
viscous working fluids, and that the Qm at high viscosity can
be a strong function of the nozzle diameter (D) instead of the
liquid conductivity (r).
Among the intrinsic properties of the working fluids (l,
q, c, e, and r), only viscosity and conductivity can be varied
by a few orders of magnitude without significantly affecting
other fluid properties. These properties and the nozzle diameter were systematically varied and the corresponding minimum flow rates measured. (Note that permittivity can be
changed by about an order of magnitude within the polarliquid limit,14,47,48 but is not systematically varied here.) As
shown in Table I, the working fluids were ethanol, ethylene
glycol or glycerol. These Newtonian working fluids spanned
three orders of magnitude in terms of viscosity. The liquids
were doped with potassium chloride to a desired range of
conductivities and confirmed with a conductivity meter
(Oakton 510). The conductivity was kept above the empirical
threshold of 1 lS/cm for high-conductivity fluids.4 To prevent subconscious bias, the exact conductivity in most
experiments was measured immediately after the measurements of the minimum flow rate (see Table S149). Note that
all three working fluids are commonly used for mechanistic
studies of cone-jets.50
To test the scaling laws, we focused on Taylor cone-jets
issued from a nozzle electrified with respect to a conducting
plate, where the working fluid was supplied at a constant
flow rate by a syringe pump (Legato 180). In addition to its
widespread adoption, the nozzle-to-plate configuration was
the easiest for accurate Qm measurements. Cone-jets were
produced on glass nozzles with outer diameters (D) ranging
104, 024103-1
C 2014 AIP Publishing LLC
V
024103-2
W. J. Scheideler and C.-H. Chen
Appl. Phys. Lett. 104, 024103 (2014)
TABLE I. Material properties of the working fluids based on literature values at 20 C. The concentration of the glycerol-water mixtures is by weight.
Working fluid
Ethanol
Ethylene glycol
90% glycerol
96% glycerol
Glycerol
l ½103 Pa s
q ½103 kg=m3 c ½103 N=m
e=e0 [–]
1.2
21
220
620
1400
0.79
1.11
1.24
1.25
1.26
22
48
65
64
63
25
41
47
44
41
from 125 lm to 840 lm at the nozzle exit. For smaller diameters, tapered fused-silica nozzles (New Objective PicoTips)
were used with outer/inner diameters of 125/75 lm,
150/100 lm, and 210/150 lm, respectively. For larger diameters, borosilicate capillaries (Vitrocom Vitrotubes) were used
with outer/inner diameters of 250/150 lm, 330/200 lm,
550/400 lm, and 840/600 lm, respectively. To assess the
potential effects of nozzle conductance, 460/260 lm stainless
steel capillaries (Hamilton 91026) were also used as
received. All glass nozzles were cut to a uniform length of
20 mm and fixed via plastic tubing sleeves to a metallic
union, through which the nozzle was grounded. The counter
electrode was negatively electrified (Trek 610E) and located
at a fixed distance of 1.5 mm from the nozzle tip. A textured
silicon substrate was used as the counter electrode to prevent
accumulation of the working fluid.51 All nozzles were oriented horizontally so that potential gravitational effects were
discernable by an asymmetric cone.
Our experimental procedures to extract the minimum
flow rate followed the methodologies in Refs. 3, 18, and 52,
and were detailed in a prior publication.19 The minimum
flow rate (Qm) required to sustain a steady jet was extracted
as the lowest flow rate bounding the V– Q stability window;
see details in supplemental Fig. S1.49 For highly viscous
working fluids containing glycerol, extra care was taken to
thoroughly flush the electrohydrodynamic systems to avoid
trapping air bubbles, and short plastic tubing connection
between the nozzle and the pump was adopted to minimize
flow capacitances. The uncertainty in the Qm measurements
was typically within 630%, as shown below in Fig. 3 for
selected conditions. All error bars were based on 95% confidence intervals.
Representative images of steady cone-jets of ethylene
glycol and glycerol are shown in Fig. 1. The steady cone-jet
of ethanol was similar to that of ethylene glycol. The main
qualitative difference between the ethylene glycol cone-jet
in Fig. 1(a) and glycerol cone-jet in Fig. 1(b) was in the
FIG. 1. Electrohydrodynamic cone-jets for (a) ethylene glycol and (b) glycerol, each at 10 lS/cm with an outer diameter of 150 lm at the exit of the
tapered glass nozzle. The cone-jet transition was localized around the conical apex for ethylene glycol with a relatively low viscosity, but spanned the
entire cone for glycerol with a relatively high viscosity.
transitional region, which bridged the Taylor cone of a
dimension dictated by the nozzle diameter with a much thinner jet (that accelerated further downstream). In the ethylene
glycol case with relatively low viscosity, the cone-jet transition was localized to within a few jet diameters; In the glycerol case with relatively high viscosity, the transition region
extends much wider with a size comparable to the nozzle
diameter.
The minimum flow rates were measured for three sets of
working fluids with wide-ranging viscosities. In Fig. 2(a),
the nozzle diameter was fixed at 150 lm and the conductivity
was varied from 1 lS/cm to 25 lS/cm for each liquid. The
minimum flow rate was a strong function of conductivity for
ethanol and ethylene glycol, but independent of conductivity
for glycerol. In Fig. 2(b), the conductivity was fixed at
5 lS/cm, and the nozzle diameter was varied from 150 lm to
550 lm. The minimum flow rate was essentially independent
of nozzle diameter at low viscosities, but became a strong
function of diameter for highly viscous glycerol.
The minimum flow rate of ethanol and ethylene glycol
in Fig. 2 exhibited trends consistent with the conventional
scaling Eq. (1). However, the parametric dependence of
glycerol contradicted Eq. (1), with the minimum flow rate independent of the electrical conductivity but dependent on the
nozzle diameter. The diameter-dependence for glycerol was
consistent with the qualitative observation in Fig. 1(b),
where the length scale of the cone-jet transitional region was
comparable to the nozzle diameter. To better understand
these scaling trends, the quantitative dependence on conductivity and nozzle diameter was evaluated in Figs. 3 and 4.
FIG. 2. The minimum flow rate (Qm) at increasing viscosity (l). (a) With a
given nozzle outer diameter of 150 lm, the Qm was dependent on conductivity for ethanol and ethylene glycol with relatively low viscosities, but not dependent on conductivity for glycerol with a much higher viscosity. (b) With
a given conductivity of 5 lS/cm, the Qm did not depend on the nozzle diameter (D) at low viscosities but exhibited a strong dependence for highly viscous glycerol.
024103-3
W. J. Scheideler and C.-H. Chen
Appl. Phys. Lett. 104, 024103 (2014)
FIG. 3. In the low-viscosity limit, the minimum flow rate (Qm) was strongly
dependent on the liquid conductivity (r) and nearly independent of the
outer diameter of the nozzle (D). The dashed line was a power law fit with
Qm r0:5660:07 for all ethylene glycol data.
FIG. 4. In the high-viscosity limit, the minimum flow rate (Qm) was strongly
dependent on the nozzle diameter (D) but nearly independent of the liquid
conductivity (r). The dashed line was a power law fit with Qm D1:4560:18
for all glycerol data except the 840 lm cases.
In Fig. 3, the conductivity of ethylene glycol was varied
by two orders of magnitude, and the resulting minimum flow
rate followed the scaling trend in Eq. (1). However, the
power-law dependence for ethylene glycol was notably
weaker with Qm r0:5660:07 , consistent with the observation in Ref. 19. Interestingly, ethanol followed a similar
trend as ethylene glycol, with an arguably larger power law
coefficient. In contrast to the strong dependence on liquid
conductivity, the minimum flow rate did not show any
obvious dependence on the nozzle diameter (Fig. 3 inset).
The measurement was also insensitive to the choice of conducting versus insulating nozzles. The qualitative trend of
decreasing minimum flow rate at increasing conductivity is
consistent with prior reports.4,14–16
In sharp contrast to the low-viscosity cases in Fig. 3, the
minimum flow rate of glycerol in Fig. 4 was essentially independent of the liquid conductivity, but depended strongly on
the nozzle diameter. For glycerol with the outer diameter
ranging from 125 lm to 550 lm, the power law was extracted
as Qm D1:4560:18 . Because of experimental limits, the nozzle diameter could be neither much smaller nor much larger.
For smaller nozzles with a diameter well below 100 lm, it
was extremely difficult to consistently pump a highly viscous
liquid like glycerol through the nozzle. For larger nozzles
with a diameter well over 500 lm, gravitational effects could
no longer be neglected; In addition, a larger nozzle diameter
could interfere with the constant nozzle-to-plate separation of
1.5 mm. With the limited range of nozzle diameters, it was
difficult to test if the Qm measurements correlated more
strongly with the inner or outer diameter, and the wetted diameter (outer diameter here) was adopted in all data analysis
to be consistent with prior studies.20,24,25 Similar to pure glycerol, the minimum flow rate of 96% glycerol exhibited a
strong dependence on nozzle diameter (Fig. 4) but nearly no
dependence on liquid conductivity (Fig. 4 inset).
The drastically different scaling regimes of the minimum flow rate in Figs. 3 and 4 are directly related to differing liquid viscosities, consistent with the effects of viscosity
observed in Fig. 2. The different scaling regimes can be tentatively rationalized with the following analysis on the
governing time scales. The flow rate scaling in Eq. (1) is proportional to the charge relaxation time, te ¼ e=r, indicating
that te governs the electrohydrodynamic processes at low viscosities. With increasing viscosity, the viscous diffusion
time which scales as tv ¼ qD2 =l may become comparable or
smaller than the charge relaxation time. When the rapid viscous diffusion process dominates over the charge relaxation
process, the minimum flow rate scaling will adopt an alternative form
Qm cD2
;
l
(2)
which can be obtained by replacing te with tv in Eq. (1). Note
that with the parametric space explored here for pure glycerol
as well as glycerol mixtures (see supplemental
Table
S149),
pffiffiffiffiffiffiffiffi
ffi
the Ohnesorge number defined as Oh ¼ l= qcD is consistently greater than 1, which indicates that viscous effects will
dominate over inertial effects and the liquid density (q) drops
out of the scaling. The scaling law in Eq. (2) can also be
viewed as the capillary-viscous velocity ðc=lÞ multiplied by
a characteristic cross-sectional area, which scales as D2 but is
likely smaller in magnitude as in Fig. 1(b).
With the caveat of the somewhat limited parametric
range, particularly the narrow range of nozzle diameters, our
existing experimental data on highly viscous working fluids
were consistent with the capillary-viscous scaling Eq. (2).
The power law dependence on the nozzle diameter extracted
from Fig. 4 appeared to be close to the Qm D2 scaling.
The independence on conductivity was observed for pure
glycerol as well as 96% glycerol. The scaling dependence on
viscosity was harder to test, given the difficulty in obtaining
Newtonian liquids with a viscosity higher than that of glycerol. Although binary mixtures were not ideal working fluids
with potential complications such as demixing at high electric fields53,54 and preferential solvent evaporation, preliminary tests with 100%, 96%, and 90% glycerol did indicate
increasing minimum flow rate with decreasing viscosity,
albeit much weaker than the l1 power law; see
for example, the three cases with 10 lS/cm and 150 lm in
024103-4
W. J. Scheideler and C.-H. Chen
Table S1.49 It should be noted that the minimum flow rate of
90% glycerol was no longer completely independent of conductivity (see Table S149), so Eq. (2) may not be fully applicable at this viscosity.
We should stress that the capillary-viscous scaling in
Eq. (2) is one of the many possible scaling relations yielding
a dependence on the nozzle diameter. Given the dominance
of viscosity with Oh > 1 and the experimentally established
independence on liquid conductivity, Eq. (2) appears to be a
reasonable choice on dimensional grounds. Our work may be
related to a recent study of pulsating cone-jets emitted from
dielectric films.34 In that study, the capillary-viscous velocity
ðc=lÞ appeared in the scaling laws validated for highly viscous working fluids, which satisfied tv te with their
extremely low conductivity.34 On the other hand, we shall
note that there are different views on the role of charge relaxation time (te) in cone-jet dynamics,4,10 and that the actual
viscous time scale may be much shorter than the simple scaling of tv qD2 =l. (For example, the viscous decay time is
tv/20 during drop oscillation,55 which is sometimes used as a
first approximation to cone oscillation.24,56) In future, the
capillary-viscous scaling Eq. (2) should be further assessed
with experiments over a wider parametric range and with a
full electrohydrodynamic model accounting for all the competing processes.
In summary, we experimentally studied the scaling laws
governing the minimum flow rate of steady Taylor cone-jets
issued from an electrified nozzle. Contrary to the wellknown scaling for low-viscosity liquids which is proportional to the charge relaxation time as in Eq. (1), an alternative scaling regime was identified for high-viscosity liquids
which seemed to be proportional to the viscous diffusion
time as in Eq. (2). The “charge-relaxation” scaling is
strongly dependent on the liquid conductivity, whereas the
“capillary-viscous” scaling is strongly dependent on the nozzle diameter. Our findings suggest precautions when applying existing scaling models of lowly viscous cone-jets to
highly viscous electrohydrodynamic systems, such as those
in miniaturized electrospraying and electroprinting.
This work was supported by the National Science
Foundation (Grant No. CBET-08-46705). We thank D. B.
Bober, O. M. Knio, and X. Qu for useful discussions.
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See supplementary material at http://dx.doi.org/10.1063/1.4862263 for
Figure S1—example V–Q operating windows for steady cone-jets; Table
S1—minimum flow rate measurements used in Figs. 2–4.
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