Rheol Acta (2013) 52:313–325
DOI 10.1007/s00397-013-0688-4
ORIGINAL CONTRIBUTION
Generation of inkjet droplet of non-Newtonian fluid
Hansol Yoo · Chongyoup Kim
Received: 3 June 2012 / Revised: 28 January 2013 / Accepted: 31 January 2013 / Published online: 17 February 2013
© Springer-Verlag Berlin Heidelberg 2013
Abstract In this study, the generation of inkjet droplets
of xanthan gum solutions in water–glycerin mixtures was
investigated experimentally to understand the jetting and
drop generation mechanisms of rheologically complex fluids using a drop-on-demand inkjet system based on a piezoelectric nozzle head. The ejected volume and velocity of
droplet were measured while varying the wave form of bipolar shape to the piezoelectric inkjet head, and the effects of
the rheological properties were examined. The shear properties of xanthan gum solutions were characterized for wide
ranges of shear rate and frequency by using the diffusive
wave spectroscopy microrheological method as well as the
conventional rotational rheometry. The extensional properties were measured with the capillary breakup method.
The result shows that drop generation process consists of
two independent processes of ejection and detachment. The
ejection process is found to be controlled primarily by high
or infinite shear viscosity. Elasticity can affect the flow
through the converging section of inkjet nozzle even though
the effect may not be strong. The detachment process is
controlled by extensional viscosity. Due to the strain hardening of polymers, the extensional viscosity becomes orders
of magnitude larger than the Trouton viscosities based on
the zero and infinite shear viscosities. The large extensional
stress retards the extension of ligament, and hence the stress
lowers the flight speed of the ligament head. The viscoelastic properties at the high-frequency regime do not appear
to be directly related to the drop generation process even
though it can affect the extensional properties.
H. Yoo · C. Kim ()
Department of Chemical and Biological Engineering,
Korea University, Anam-dong, Sungbuk-ku, Seoul 136–713,
South Korea
e-mail: [email protected]
Keywords Drop-on-demand inkjet · Elasticity · Shear
thinning · Infinite shear viscosity · Jeffery–Hamel flow ·
Strain hardening · DWS microrheology
Introduction
As the inkjet printing technology has widen its application
to bio and electronic industries beyond household or office
inkjet printers (Basaran 2002; Schubert 2005; de Gans et al.
2004), many different kinds of inks have to be handled. In
most cases, inks are suspensions or polymeric liquids (de
Gans et al. 2005), and hence most of the inks are rheologically complex fluids showing shear-dependent viscosities
and/or elastic characteristics. Some additives such as surfactants are usually added in suspensions for stabilization
and better performances. This can make the rheology of
suspension more complex. However, inks have not been
characterized properly especially at the operating conditions
of inkjet printing, and the processing conditions have been
sought mostly through trial and error basis.
To generate inkjet droplets, either the continuous jetting or drop-on-demand (DOD) method can be used (Derby
2010). In the DOD method, droplets are generated by applying a pressure wave to a liquid-filled nozzle. Then, a portion
of liquid is squeezed out of the nozzle overcoming the surface tension force, and the liquid element is detached from
the nozzle tip by inertia and capillary force. In this stage,
the fluid element becomes elongated before detachment,
and the elongated liquid thread is either contracted so that
a single drop is generated or divided into the leading drop
and some smaller satellite drops by instability mechanisms.
In some cases, satellite drops can be merged into the leading drop. It is known that the formation of satellite drops
should be avoided for better printing quality, and hence the
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determination of the proper window on the operating parameters for a single drop generation is one of the most important issues in inkjet droplet generation. It has been reported
that drop generation characteristics are governed by Ohnesorge number (Oh) of the drop, which is the ratio of viscous
time scale and surface tension time scale and defined as
follows (Derby 2010):
η
Oh = √
,
ρRγ
(1)
where η is viscosity, ρ is density, R is radius, and γ is
surface tension. For Newtonian fluids, Oh is unequivocally
defined since fluid properties are independent of flow conditions. But in the case of shear thinning fluids, viscosity
is a function of shear rate, and hence the operating window
cannot be predicted based on the theory for Newtonian fluids (Lai et al. 2010; Tai et al. 2008). It has been the usual
notion that the rheological behaviors at large strain rates
control the generation of inkjet droplet (Hoath et al. 2009)
since the drop generation process is an ultrahigh shear rate
process with an average shear rate of 105 s−1 order. Actually, inks show finite viscosities as shear rate goes either
to zero or a very large value. In this case, some questions
should arise naturally. Is the zero shear viscosity not relevant to the generation of inkjet drop generation? If this is so,
is the infinite shear viscosity the only variable that affects
drop generation? If not, what other properties are relevant
to drop generation? In the present paper, we have tried to
answer these questions by using a class of fluids (as model
inks) which show various rheologically complex behaviors
such as elasticity, shear thinning, and strain hardening.
Shore and Harrison (2005) reported that the presence of
a small amount of polymer in a Newtonian solvent can have
a significant change in the inkjet drop generation characteristics. Especially, satellite drop formation is suppressed and
the drop velocity is significantly lowered by the addition of
polymer. Using two different types of polymers (linear and
star polymers), de Gans et al. (2005) reported that the distance traveled by the primary droplet was dependent only
on input voltage to the piezo-element and independent of
polymer concentration, molecular weight, and topological
architecture. They also reported that the rupture of the ligament was dependent on the rheological properties of the
solution. Hoath et al. (2009) noticed that, in the generation of inkjet droplet of elastic polymer solutions, the final
main drop size was independent of polymer concentration
even though the length of the ligament increased markedly
with the elasticity of the fluid. In the meantime, Hoath
et al. (2009) did not observe any correlation between low
shear viscosity and jetting behavior for the fluids they investigated, but the jetting behavior was well correlated with
high-frequency rheological properties measured at 5 kHz
using a piezoelectric axial vibrator rheometer. Here, it is
noted that the drop generation is a highly nonlinear process
of large extension and high extensional rate, and therefore
linear viscoelastic properties may not be correlated quantitatively with the nonlinear process. Morrison and Harlen
(2010) investigated the effects of viscoelasticity numerically
on drop formation in inkjet printing by using viscoelastic
fluids represented by the single-mode FENE-CR constitutive equation (Chilcott and Rallison 1988). They showed
that the ligament became longer for elastic liquids and the
formation of satellite drops was suppressed by elasticity.
Also, they argued that the lowering of drop speed was due
to elasticity. Recently, Hoath et al. (2012) presented a quantitative model which predicted three different regimes of
behavior depending upon the jet Weissenberg number (Wi)
and extensibility of polymer molecule. They predicted Newtonian regime (Wi < 1/2), viscoelastic regime with partial
extension (1/2 < Wi < L, where L is the extensibility of
polymer chain), and fully extended regime (Wi > L). They
also gave the scaling law for the maximum polymer concentration at which a jet of a certain speed could be formed as
a function of molecular weight of polymer. Their analysis
is based on the FENE-CR model which is valid for solutions of flexible polymers. Also, their analysis is limited to
the detachment process, and hence the model cannot predict
the drop size. Therefore, more studies on drop size and drop
velocity should be still required to understand the mechanism of inkjet drop generation of viscoelastic fluids. All
of these papers argued that the elasticity has a significant
effect on the generation of inkjet droplet of elastic solution.
However, they did not give the detailed reason why elasticity could affect the drop formation. In the present paper, we
have examined the elongation characteristics and linear viscoelasticity of inks along with the flow of inks inside the
nozzle and their effects on inkjet drop generation.
The elongation of liquid thread has been an important
issue in rheology. In continuous jetting, dripping, necking, and breakup of liquid bridge, the thinning of a liquid
filament is driven by capillarity and resisted by inertia, viscosity, and elasticity. On the other hand, the stretching of
the ligament in the early stage of drop formation is primarily driven by the inertia (in the main flight direction) of
the ligament head and resisted by surface tension, inertia
(in the perpendicular direction to the main flight direction), viscosity, and elasticity of fluid. Hence, the detailed
flow should not be the same. At the final stage of filament
breakup, it is known that the breakup process is determined
by the natural variables of surface tension and fluid properties regardless of boundary and initial conditions (Eggers
1993; Renardy 2004; McKinley 2005). Therefore, we may
gain useful information from the capillary breakup test.
The visco-elasto-capillary thinning of complex fluids has
been studied extensively since Eggers (1993) first found
the similarity solution for the one-dimensional governing
Rheol Acta (2013) 52:313–325
315
equation and subsequent reports on the similarity solutions
for various non-Newtonian models. Detailed reviews on the
self-similar solutions for non-Newtonian models were given
by Renardy (2004), and comprehensive reviews were given
by McKinley (2005) on capillary thinning of liquid bridges
and its applications to extensional rheometry. The studies
on capillary thinning have shown that, in the case of Stokes
flow of a Newtonian fluid, the filament breaks off at a finite
time tc and the radius of the filament changes with time as
follows (Papageorgiou 1995):
Rmid
σ
= 0.0709
(tc − t)
R0
η s R0
(2)
Then, the extensional rate at the middle of the filament is
given as follows:
ε̇mid (t) = −
2 dRmid
2
=
Rmid dt
t − tc
(3)
In the above equations, ε̇ is extension rate, R is radius,
ηs is viscosity, and t is time. Also, subscripts mid and 0
denote the value at the midpoint of the filament and initial
radius, respectively. In the case of elastic fluid (McKinley
2005), the capillary thinning flow becomes a homogeneous
extensional flow with
Wi = λ1 ε̇ = 2/3
and the radius changes with time as follows:
GR0 1/3
Rmid
=
exp (−t/ (3λ1 ))
R0
2σ
(4)
(5)
where G is modulus and λ1 is the longest relaxation time.
In this case, there is no finite breakup time and there is
a long tail. This equation is valid for a dilute solution of
infinitely extensible polymers. But in a real polymeric liquid, polymers cannot be extended infinitely. Renardy (2002)
and Fontelos and Li (2004) have shown that, for viscoelastic fluids of Giesekus and FENE-P types, the jet diameter
decreases linearly with time when close to the breakup:
R(t)
σ
=
(6)
(tc − t)
R0
2ηE R0
This result means that, as polymer molecules are fully
stretched at a sufficiently large strain rate, the extensional
viscosity approaches a constant value and the fluid behaves
as a Newtonian fluid with a constant extensional viscosity of ηE (McKinley 2005; Stelter et al. 2002, 1999). From
this relationship, we may obtain the extensional viscosity of
inkjet fluid from the capillary breakup experiment. If we can
observe a linearly decreasing filament radius while exhibiting a cylindrical filament, ηE can be obtained from the slope.
The ηE value obtained here can be close to the true extensional viscosity at the inkjet drop generation condition since
polymer coils can be almost fully stretched at the high strain
rate, and hence ηE value approaches the limiting value.
In the present research, we have attempted to correlate
the relevant variables to each drop-generating step by performing inkjet drop generation experiments with xanthan
gum solutions together with numerical simulations on the
flow inside the nozzle. Since xanthan gum solutions are
less elastic than most of the flexible polymer solutions,
the result presented here can describe the practical inkjet
problems more realistically. In the analysis of experimental
data, we used the rheological properties at the real processing condition measured by the diffusive wave spectroscopy
microrheology method for high-frequency linear viscoelastic properties and capillary breakup method for extensional
viscosities at high strain rate as well as the conventional
rotational rheometry. The result shows that drop generation
process consists of two separate stages: At the first stage,
a certain amount of liquid is ejected from the nozzle and
the drop volume is determined by this step. Especially, it
has been found that drop volume is determined mainly by
the infinite shear viscosity of the xanthan gum solution. At
the second stage, the ejected liquid is pinched off from the
fluid inside the nozzle by the inertia of the liquid and the
pulling-back action of piezo-element, and the drop velocity
is determined by the ejection velocity and the extensional
viscosity of fluid. As xanthan gum solutions show most of
the important characteristics of non-Newtonian fluids such
as elasticity, shear thinning, and extensional thickening, the
present study can give an insight into the processing of nonNewtonian fluids for various applications by using a DOD
inkjet printing system. The present result can be also used
even for predicting the jetting behavior of the solution of a
flexible polymer which can be regarded as a different class
of fluids from the xanthan gum solution.
Experiment
To investigate the generation of inkjet drops, we set up an
inkjet system as shown in Fig. 1, which is the same as the
set that one of the authors used for the previous studies on
Fig. 1 Schematic diagram of the experimental setup
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Rheol Acta (2013) 52:313–325
spreading of inkjet drop (Son et al. 2008). In the present
case, there is no such part as the solid surface. The system
consists of an inkjet nozzle, a jetting driver (pulse generating
system), a high-speed camera, and an illumination source.
Inkjet system and imaging
The inkjet droplet was generated by a piezo-type nozzle purchased from MicroFab Co. (Model # MJ-AT). The nozzle
diameter at the exit was 50 μm. In Fig. 2, the inside geometry of the nozzle is shown. In taking the picture, the nozzle
filled with air was immersed in a decalin-filled square box.
Since decalin has the same refractive index as the glass, the
refraction at the curved nozzle surface can be avoided.
To generate droplets, a bipolar wave form was used as
shown in Fig. 3. During the rise period (a), the piezoelement expands for fluid intake from the reservoir and this
state continues during the dwell period (b). During the fall
period (c), the piezo-element shrinks and fluid is ejected out
of the nozzle. This state continues during the echo period
(d). Finally, the piezo-element expands to return to the initial
state while completing a cycle (e). Depending on the time
intervals and the voltages imposed on the piezo-element, a
drop or drops of different sizes and velocities are generated.
In the present research, the rise and fall times in the voltage pulse to the nozzle were set at 2 μs. The dwell and echo
times were in the range of 4–32 μs, and the dwell and echo
voltages were in the range of 12–50 V. We performed drop
generation experiments by varying operating conditions and
measured the drop size and velocity. All the experimental
runs were performed at the room condition. The high-speed
camera (IDT, XS-4) was triggered by the jetting driver as a
droplet was ejected from the inkjet nozzle. The camera was
equipped with a microscopic objective lens (Mitutoyo, M
Plan Apo) with the magnification of ×5. As the illumination source, a back lighting system (Stocker Yale, # 21 AC,
180 W) was installed.
The CCD camera can capture 50,000 frames per second
and the pictures were taken with this mode. The exposure
time was 1 μs. When this fast mode was used, the number
Fig. 3 Pulse wave form for generating inkjet droplets: a expansion of
nozzle chamber, b delay for pressure wave propagation, c compression of fluid for ejection, d delay for pressure wave propagation, and e
nozzle chamber expansion to the initial state
of pixels per frame has to be small (512 × 48 pixels). The
pixel size was 3.76 μm. One may use the flash videography
method to get the better quality as demonstrated by van Dam
and Le Clerc (2004) and Dong et al. (2006). But, in the case
of non-Newtonian drops, the reproducibility of drop generation was not as good as in case of Newtonian fluids; hence,
we decided not to use the flash videography method used in
the literature. Considering that the drop diameter and travel
distance are in the order of 50 μm and 1 mm, respectively,
the numbers of pixels to cover these sizes are about 15 and
150. Therefore, the image resolution was enough to measure
drop diameter and velocity.
Materials
Three kinds of Newtonian fluids were prepared with different viscosities by mixing deionized water and glycerin
(Sigma-Aldrich Co.). Shear-thinning fluids were prepared
by dissolving xanthan gum (Sigma-Aldrich Co.) in deionized water or in one of the water–glycerin mixtures. Table 1
shows the composition of the fluids tested here. The
shear viscosity and linear viscoelastic properties of liquids
were measured by a rotational rheometer with a Couette
fixture (AR2000, TA Instrument). High-frequency linear
Table 1 Newtonian base fluids
Fig. 2 Inside geometry of the nozzle. The inlet diameter is 456 μm
and the exit diameter is 50 μm
Sample name
1 cP
4.5 cP
10 cP
16.5 cP
Deionized water, wt%
Glycerin, wt%
Viscosity, mPa s
Surface tension, mN/m
100
0
1
71.6
55
45
4.5
66.0
40
60
10
67.5
32
68
16.5
66.9
Rheol Acta (2013) 52:313–325
317
viscoelastic properties were measured by a diffusive wave
spectroscopy (DWS) microrheology rheometer (RheoLab,
LS Instrument) using polystyrene spheres of 520 nm in
diameter as tracer particles. The cuvette thickness was 2 mm
and the properties were measured under the transmission
mode. The extensional viscosity was estimated by using a
capillary breakup apparatus (CaBER, ThermoHaake Co.).
To handle low-viscosity fluids, two small plates of diameter
2 mm were machined from titanium. The initial gap distance was the same as the radius of the plate and the initial
deformation was imposed to 2.5 mm for 20 ms. The surface
tension was measured by using the Du Noüy ring method
(K9 Tensiometer, KRÜSS GmbH).
Result and discussion
Figure 4 shows the shear viscosities of two sets of fluids
tested in the present research. The mixtures of DI water and
glycerin have shear-independent viscosities while all xanthan solutions have shear-thinning viscosities. The viscosity
of xanthan solution is fitted to the Carreau model (Bird et al.
1987):
η − η∞
=
η0 − η∞
1
1 + (β γ̇ )2
(7)
,
1−n
2
and the model parameters are listed in Table 2. In the table
(also in Fig. 4), we note that the infinite shear viscosities
of xanthan gum solutions of the same base solvent are only
slightly changed from or almost the same as the viscosity
Table 2 Xanthan gum
solutions studied here and their
Carreau model parameters and
surface tension
Fig. 4 Viscosities of xanthan gum solutions for differing solvents and
xanthan gum concentrations. The symbols are measured values. The
solid lines are the Carreau model fit
of the solvent. In the following discussion, we will compare the drop generation characteristics of the fluids with
the same base fluid systematically. Figure 5 shows linear
viscoelastic properties of some of the fluids listed in Table
2. Figure 5 shows that the shapes of G and G for the
xanthan gum solutions are not similar to those of polymer solutions of flexible polymers such as Boger fluids in
that the typical slope of 2 for G at low-frequency regime
does not appear yet at the lowest frequency of 0.1 s−1 .
The data obtained from the DWS microrheology appear to
be reasonably extended to the conventional rheometry data.
1.0 cP
η0 (mPa s)
η∞ (mPa s)
β, s−1
n
σ (mN/m)
4.5 cP
50 ppm
100 ppm
200 ppm
50 ppm
100 ppm
200 ppm
2.2
1.2
0.25
0.63
71.4
3.9
1.4
0.22
0.38
71.6
7.09
1.6
0.24
0.52
71.4
6.89
4.7
0.28
0.61
65.7
12.6
4.0
0.47
0.71
66.5
28.8
4.3
0.68
0.41
66.0
10 cP
η0 (mPa s)
η∞ (mPa s)
β, s−1
n
σ (mN/m)
16.5 cP
50 ppm
100 ppm
200 ppm
50 ppm
100 ppm
200 ppm
17.2
9.3
0.77
0.73
67.3
29.4
8.7
1.30
0.72
67.2
71.8
10.6
1.85
0.61
67.4
29.3
16.0
1.66
0.75
66.8
48.7
16.5
0.95
0.64
67.2
136.2
14.0
10.3
0.69
66.9
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Fig. 5 Viscoelastic properties of some xanthan gum solutions: a
10-cP-based solutions, b 16.5-cP-based solutions. Symbols were
obtained by rotational rheometry. Solid lines were obtained by DWS
microrheology
One important characteristic is that the xanthan gum solutions have smaller G than G for all the frequency regimes
tested here including the DWS microrheology measurement
regime. This implies that the elastic effect should not be
large even at the inkjet drop generation condition (shear rate
of 105 s−1 ).
Figure 6 shows some typical drop generation patterns for
differing fluids at differing conditions. In Fig. 6a, a thin
liquid ligament with a spherical head is formed, and then
the ligament tail contracts to the head eventually. Therefore,
only one drop is generated. In Fig. 6b, the situation is much
the same as Fig. 6a, but the tail is separated from the main
head and becomes a satellite drop. More than one satellite
drop can be generated and these drops can be coalesced
depending upon the relative velocity between the main and
satellite droplets. The separation of the satellite drop is
caused by the capillary instability or end pinching (Stone
Fig. 6 Typical drop generation patterns. The time interval between
two adjacent frames is 20 μs. a Single-drop formation for the 100-ppm
solution in 4.5-cP solvent. The tail is shrunk to the main drop. Driving
voltage is 30 V. b A satellite drop generated by the separation from the
tail for the 50-ppm solution in 10-cP solvent. Driving voltage is 36 V. c
A satellite drop generated by reflected acoustic waves for the 50-ppm
solution in water. In this case, the tail is shrunk to the main drop, but a
new drop is generated behind the main drop
1994; Stone et al. 1986). These drops may be called “satellites from tail.” In Fig. 6c, a satellite drop is generated by
a different mechanism. In this case, the satellite drop is not
linked to the original ligament and appears to be generated
by the reflected acoustic wave inside the nozzle. These satellite drops may be called “satellites from reflected wave.”
In the following discussion, we will consider the cases for
which only one single drop is generated. To generate one
single drop without satellites, a proper operating window
Rheol Acta (2013) 52:313–325
for voltage and time of dwell and echo has to be chosen as
well as rise and fall time. Since there are too many combinations for operating parameters, we confine ourselves to
the following cases. First, the rise and fall times were fixed
at 2 μs. Dwell and echo voltages were set to be equal, and
then we changed dwell/echo time to find out the condition
at which the drop velocity became the maximum value. In
most of the cases, the drop velocity became the maximum
or close to the maximum value when dwell/echo time was
24 μs. Therefore, we fixed dwell/echo time at 24 μs. The
optimum dwell/echo time (the condition at which the drop
velocity attains the maximum value) is closely related to the
length of the nozzle and acoustic velocity of fluid (Bogy
and Talke 1984). Since the nozzle length is fixed and the
acoustic velocity is not much different (between 1,481 m
s−1 (water) and 1,980 m s−1 (100 % glycerin) at 20 ◦ C),
the optimum dwell/echo time appears to have similar values. After choosing the dwell/echo and rise/fall times, we
319
performed experiments while varying dwell/echo voltage
and collected data when only a single drop was generated.
Figure 7 shows the drop velocity variations for the fluids listed in Table 1. First, we note that, for all cases,
as dwell/echo voltage (voltage hereafter) increases, drop
velocity increases. Also, the minimum voltage required to
generate a single drop increases with xanthan gum concentration. At a fixed voltage value, drop velocity becomes
smaller with an increase in xanthan gum concentration. For
example, when the viscosity of solvent is 4.5 cP, the drop
velocity is 4.5, 3.2, and 0.6 m s−1 when the driving voltage
is 24 V and the xanthan gum concentration is 0, 50, and 100
ppm, respectively. At 24 V, the 200-ppm solution could not
be ejected due to the strong pull back of the ligament to the
nozzle. Similar patterns are observed for all cases shown in
Fig. 7. Figure 8 shows the drop volume changes with driving voltage. Even though we have only a limited number
of data points for each fluid with the same base fluid, we
Fig. 7 Velocity of drop for differing driving voltage. The solvent is a water, b 4.5 cP, c 10 cP, and d 16.5 cP
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Fig. 8 Total ejected volume for differing driving voltage. The solvent is a water, b 4.5 cP, c 10 cP, and d 16.5 cP
can note that the drop volume is a function of driving voltage only regardless of xanthan gum concentration since the
data points are continued along a single curve (with some
overlaps) for solutions from the same base fluid except for
the 200-ppm solutions. In the case of the 200-ppm solutions in 16.5-cP solvent, it was impossible to generate a
drop for the full range of driving voltage. The concentration independency was also reported by Hoath et al. (2009)
(the deviation of the 200-ppm solutions from the general
trend will be considered later in this report). The independency of concentration on drop volume indicates that drop
volume is a function of infinite shear viscosity only. This
is quite in contrast to the fact that drop velocity is a strong
function of xanthan gum concentration. The two contrasting
observations imply that drop generation process consists of
two separate stages, and each stage is governed by different physical properties: At the first stage, a certain amount
of liquid is ejected from the nozzle and the drop volume
is mainly determined by this step. At the second stage, the
ejected liquid is pinched off from the fluid in the nozzle by
the inertia of the liquid and the pull-back action of the piezoelement, and hence the drop velocity is determined by the
ejection velocity at the nozzle tip and the extensional characteristics of fluid. In the next discussion, we have described
these two steps sequentially.
To understand the fluid ejection step, we need to know
the amount of fluid that is ejected by the acoustic pressure wave and its relation with fluid properties. Since it
is very difficult to measure the velocity profile at the exit
accurately, we estimated it by numerical simulation using
Fluent™, a commercial software package based on the finite
volume method. We used 6,320 elements and 6,657 nodes
in the simulation. The operating conditions were obtained
by the following method. From the two consecutive images
on jetting of a Newtonian fluid, the flow rate through the
nozzle was obtained for a typical pressure wave form. From
the numerical simulation results, we read the pressure drop
between the manifold and the exit of the nozzle that gave
the same flow rate as the experimentally observed flow
rate value for the Newtonian fluid. We assumed that the
Rheol Acta (2013) 52:313–325
Fig. 9 Velocity profiles of Newtonian fluids at the nozzle exit at a
typical pressure difference of 20,000 Pa between the nozzle inlet and
exit
acoustic velocity of a dilute polymer solution was the same
as the solvent. Then, the pressure drop between the manifold and the exit should be the same regardless of fluid
for the given wave form. Using the same pressure drop, we
calculated the velocity profile while varying fluids of different zero shear viscosities. Since xanthan gum solutions
show elasticity, it has to be considered in velocity profile
calculation for the flow through the converging geometry
of nozzle due to Lagrangian unsteadiness. Until now, the
converging flow of elastic fluids has been treated only for
Oldroyd B and upper convected Maxwell fluids (Hull 1981;
Evans and Hagen 2008). Since the fully nonlinear constitutive model for xanthan gum solutions is not available at the
present time, we were not able to include the elastic effect
in the numerical simulation properly. Rather, we neglected
the elasticity of xanthan gum solutions since the viscoelastic measurements showed that G was substantially smaller
than G even at high-frequency regimes. Therefore, even
though the velocity profile obtained by the present method
may not represent the physics of the problem exactly, we
can obtain at least semiquantitative result. The effect of
elasticity will be considered later in this study. Figure 9
shows the velocity profiles of Newtonian fluids of differing viscosities for a typical pressure difference of 20,000
Pa between the exit and the entrance of nozzle. When viscosity is 1 cP, the exit velocity is severely blunted. The
blunting of velocity profile is caused by the inertial effect
in the Jeffery–Hamel flow (Batchelor 1967). In the blunted
r
region, the inertial term ρ vr ∂v
∂r dominates over the viscous
term and is balanced with the pressure term, while near the
solid boundary, the viscous term dominates over the inertial term. The blunted velocity profile at the nozzle exit can
be surmised from the experiment for a low-viscosity fluid
(please see the third frame of Fig. 6a). Dong et al. (2006)
321
observed the velocity blunting of water more clearly. As viscosity increases, the velocity profile becomes close to the
parabolic profile. Next, we performed the simulation for the
Carreau model fluids with the experimentally fitted numerical parameters. In Fig. 10, we have plotted the velocity
profiles for two different sets of fluids with the same infinite shear viscosities. It is seen that the velocity profiles are
almost the same regardless of zero shear viscosity of fluids
tested here, in other words, xanthan gum concentration. This
is because at the central region, the inertial term is dominant
while shear rate is extremely large near the boundary as in
the case of a Newtonian fluid. Therefore, the shear rates at
which viscosity changes appreciably occur for a very narrow
region near the center. This result means that the exit velocity is almost the same regardless of zero shear viscosity as
long as infinite shear viscosities are the same for differing
fluids. The dependence of drop size only on driving voltage
Fig. 10 Numerical simulation result on the effect of xanthan gum concentration on the velocity profile at the nozzle exit for different solvent:
a water and b 16.5 cP. In each case, the driving voltage is different to
simulate real situations
322
has been already observed in the experiment as described
above. Therefore, we can argue that drop volume is mainly
determined by infinite shear viscosity. Also, from the match
between experimental and simulation results on drop size,
we may argue that elasticity is not important in the present
case except for the 200-ppm solution. Experimentally, the
ejected volume of the 200-ppm solutions is smaller than the
pure solvent. It appears that the difference in experimental
results is due to the elasticity of fluids which has not been
taken into account in the Carreau model. This point will be
considered more in this paper.
Next, we consider the detachment of a drop. During the
detachment process, the ejected fluid is elongated and the
neck becomes thinned. From the images shown in Fig. 6, we
can estimate the order of the extension rate. Knowing that
the frame rate is 50,000 s−1 and the ligament length changes
from 0 to 400 μm between four frames, the average extensional rate ((
L/
t)/Laverage ) is 25,000 s−1 . As far as the
authors are aware of, the extensional viscosity at this high
Rheol Acta (2013) 52:313–325
Fig. 12 Filament shapes during thinning for the 100-ppm xanthan
gum solution in 16.5-cP solvent. From the left: after the loading, just
after the pulling apart, establishment of the cylindrical shape, and just
before the breakup
extensional rate cannot be measured by a commercial extensional rheometer as of now. Therefore, we used the CaBER
(ThermoHaake Co.) to estimate the extensional viscosity
based on the liquid bridge stretching. In Fig. 11, we have
shown the changes in the diameter of the stretched liquid
bridge as a function of time for some of the samples tested in
the present study. In Fig. 11, we note that the time evolution
of thread diameter is substantially delayed when xanthan
gum concentration increases, meaning that the extensional
viscosity increases with the increase in xanthan gum concentration. Just before the breakup, the diameter decrease
pattern changes to an exponential shape. It appears that, just
before breakup, the dominant resistance to filament thinning
is changed to inertia. In the figure, we note that the diameter
changes become linear after a transient period and until they
become exponential. We find that the filament has a cylindrical shape during the thinning process as shown in Fig. 12.
From the cylindrical filament shape, we can confirm that
the thinning process follows the elasto-capillary thinning
regime. Also, from the linear decrease in R(t) with time, we
can confirm that the extensional viscosity is the same during
the linear decrease. Hence, we can calculate the extensional
viscosity and extensional rate by using Eqs. 6 and 3, respectively. In Table 3, we have listed extensional viscosities and
extensional rates obtained by this method. In all cases, the
xanthan gum solutions show much larger extensional viscosities compared with the corresponding solvents. In the
Table 3 Extensional viscosity of some selected samples
Fig. 11 Typical results on the diameter change of liquid filament in
the capillary breakup experiment. The piston movement stopped at t =
0; a 10-cP-based solutions and b 16.5-cP-based solutions
Sample
Range of ε̇, s−1
ηE , Pa s
10 cP + 100 ppm
10 cP + 200 ppm
16.5 cP + 100 ppm
16.5 cP + 200 ppm
108 ± 17–256 ± 112
47 ± 14–142 ± 83
87 ± 26–172 ± 96
29 ± 9–58 ± 34
4.2 ± 0.6
6.3 ± 1.1
4.0 ± 0.6
9.0 ± 2.0
For each sample, more than seven runs were averaged
Rheol Acta (2013) 52:313–325
table, in the case of the 200-ppm xanthan gum solution in
10-cP solvent, the extensional viscosity is 6.3 Pa s for extensional rates between approximately 47 and 142 s−1 . The
extensional viscosity of 6.3 Pa s is much larger than the
Trouton viscosity (three times the shear viscosity) based on
the zero shear viscosity of 215 mPa s and the Trouton viscosity based on the infinite shear viscosity of 30 mPa s. The
substantially larger extensional viscosity than either of the
Trouton viscosities is due to the strain-hardening behavior
of the xanthan gum polymer. Strain hardening occurs when
polymers are stretched and aligned. The degree of alignment
should be much larger for xanthan gum (stiff polymer) than
that for flexible polymers. The easiness of alignment can be
seen from the strong shear-thinning viscosity. This means
that the extensional viscosity obtained in this study can be
quite close to the true value at the drop generation condition. The true values can be larger than the values measured
here, but the measured values can be useful enough to confirm that the extensional stress reduces the velocity of the
inkjet droplet.
Since the extensional rate at the pinch-off condition of
the drop generation process (in the order of 25,000 s−1 as
described above) is much larger than the value at the measuring condition of 47–140 s−1 , the extensional viscosity
at the pinch-off condition should not be smaller than 6.3
Pa s considering the extensional hardening characteristics
of polymers. The value 6.3 Pa s in the present case may be
the value at the fully extended state and strain hardening is
already saturated. If this is the case, the extensional viscosity at the pinch-off condition will be at least 6.3 Pa s. Here,
one may raise a question whether the polymers inside the
ligament are fully extended, and hence the extensional viscosity obtained by CaBER can be applied to the ligament
stretching. To warrant the application, we have compared
two strains as follows: First, the Hencky strain of the filament from the unstretched state in the CaBER experiment
is
D02
l(t)
ε(t) = ln
= ln
,
(8)
l0
D(t)2
where l and D are the length and diameter of the filament at time t and the subscript 0 denotes the value before
stretching. Next, the total strain of the inkjet ligament can
be estimated as follows: Before the breakup from the nozzle, the ejected liquid element can be divided into two parts:
drop head and ligament. The volume of the ligament (Vl )
can be calculated from the difference between the drop volume (Vd ) and the drop head volume (Vh ): Vl = Vd − Vh .
The ligament volume Vl is assumed to be maintained. When
liquid comes out of the nozzle, the diameter of the liquid
element is the same as the nozzle diameter (d), and hence
the liquid volume is extended from a cylinder of length
323
Vl / π d 2 /4 to a cylinder of length l(t). Then, the Hencky
strain is given as follows:
2
π d l(t)
εl (t) = ln
(9)
4Vl
In estimating the strain of the filament, the drop with the
lowest velocity we observed is used for a conservative estimate. For drops with higher velocities, the strain rate will be
higher. In the case of the 100-ppm solution in 16.5-cP solvent, the Hencky strain when the ligament length becomes
240 μm just before the breakup is ln 20 while the strain of
the filament during the CaBER experiment is ln 16 when
the filament diameter begins to decrease linearly (0.02 s in
Fig. 11a). At another case of the 200-ppm solution in 10-cP
solvent, when the ligament length is 214 μm, these values
are ln 33 and ln 16, respectively. Considering that the strain
rate is much higher for the ligament stretching during drop
generation than the filament stretching during the CaBER
experiment, one can confirm that the polymers in the ligament stretching are almost fully stretched, and therefore the
extensional viscosity obtained by CaBER can be applied to
the ligament stretching.
As the extensional viscosity of the fluid thread is much
larger than that of the Newtonian fluid with the same infinite shear viscosity, during the extension of the thread, the
extensional stress will strongly retard the deformation or
breakup. Also, before the pinch off, the extensional stress
retards the flight of the drop head. Therefore, even if drop
sizes are the same regardless of concentration of xanthan
gum as long as the infinite shear viscosity is the same, flight
velocity is strongly dependent on xanthan gum concentration. Due to the limited spatial resolution of the image and
time interval between two subsequent frames, we have not
been able to estimate the force acting on the head quantitatively to estimate the velocity change. As a rough estimate,
πD 3 v
the inertial force acting on the drop head is m dv
dt ≈ ρ 6 t
(m and v are the mass and the velocity of the ligament
head, respectively) while the force due to extensional stress
2
is π Rneck
× ηE ε̇. If we insert a set of numerical values at
−2
30-V driving voltage ( v
t = 50,000 m s , D = 50 μm,
Rneck = 5 μm, ηE = 6.3 Pa s, ε̇ = 25,000 s−1 ), the
inertial and extensional forces are 6 × 10−6 and 1 × 10−5
N, respectively. In this case, the drop velocity is substantially decreased from the ejection velocity at the nozzle tip,
and hence it should be impossible to detach the ligament
from the nozzle. In this case, the surface tension force is
π Dσ ∼
= 1.0 × 10−6 N and is almost 1 order smaller than
the inertia or extensional force. Hence, the surface tension
force does not strongly retard the deformation. As shown in
Fig. 7c, for the 200-ppm solution, no drop is generated at
this condition of 30 V and the minimum driving voltage is
36 V. For the case of the 100-ppm solution, the extensional
viscosity is 4.2 Pa s and the extensional force is 6 × 10−6
324
N. In this case, the inertial force and the extensional force
are balanced and the drop is barely generated as shown in
Fig. 7c. Of course, this calculation is just for an example,
and depending on diameter and other factors, there should
be wide variations in forces.
Following the argument of Hoath et al. (2009), we have
checked whether viscoelastic properties at high-frequency
regime are correlated with the drop
generation characteristics. In the present case, the G |G∗ | values at 5 kHz
(31,400 s−1 ) are almost the same for two different fluids in Fig. 5a, b, and hence the correlation could not be
found. This implies that the linear viscoelastic property cannot affect the drop generation process directly and the effect
of added polymers manifests itself as increased extensional
viscosity which primarily affects the detachment process.
The increased extensional viscosity may play a role also in
the converging flow inside the nozzle at high xanthan gum
concentrations, too. As seen in Fig. 8, the drop size of the
200-ppm solutions is smaller than those from less concentrated solutions. This difference appears to be caused by
the increase in extensional viscosity of the 200-ppm xanthan gum solutions inside the nozzle. This is because the
flow through the conical region of the nozzle is strongly
extensional, especially at the central region. However, it is
expected that the contribution of extensional stress may not
be as large as in the case of ligament extension. This is
because the mean residence time of polymer chains within
the region is very short, and hence the polymer chains may
not be fully extended. It can be confirmed from the fact
that the reduction of ejection velocity is not noticeable for
less concentrated solutions. We are sure that the analysis of
the detailed process should be performed through an elaborate constitutive modeling on the xanthan gum solutions
and possibly numerical solutions of the governing equations based on the model. Another issue is the location of
detachment. Depending on extensional properties, the location of detachment can vary, which results in the droplet size
change.
Summary and conclusion
In this study, the generation of inkjet droplets of nonNewtonian fluids has been investigated experimentally.
Noting that most of inks used in inkjet technology are rheologically complex fluids, xanthan gum solutions in water–
glycerin mixtures have been chosen as model inks. The
rheological properties of xanthan gum solutions have been
characterized by the diffusive wave spectroscopy microrheological method for high-frequency viscoelastic properties
and the capillary breakup method for extensional viscosity
as well as conventional rotational rheometry for viscosity
and linear viscoelastic properties for moderate values of
Rheol Acta (2013) 52:313–325
shear rate or frequency. The result shows that drop generation process consists of two independent processes of
ejection and detachment. The ejection process is found to
be controlled primarily by high or infinite shear viscosity.
Elasticity can affect the flow rate (drop size) through the
converging section of an inkjet nozzle. However, the elastic effect may not strongly affect the flow rate since the
residence time of polymer molecules of the section is too
short for the elastic stress to grow to a significant level. The
detachment process is controlled by the extensional viscosity. Due to the strain hardening of polymers, the extensional
viscosity becomes orders of magnitude larger than the Trouton viscosities based on zero and infinite shear viscosities.
The large extensional stress retards the extension of ligament and hence lowers the flight speed of the ligament head.
The viscoelastic properties at high-frequency regime do not
appear to be directly related to the drop generation process even though it is surmised that the elastic effect should
strongly affect the extensional properties.
We have not performed the numerical simulation on the
whole drop generation process since, first of all, there exists
no constitutive relation which is reasonably well matched
to the experimental data for xanthan solutions. Also, the
detailed numerical analysis is far beyond the scope of the
present paper. But it should be worth doing by considering
the flow inside the nozzle and detachment process together.
This is especially important in non-Newtonian liquids since
polymers are elongated during the converging flow inside
the nozzle and the elongated polymers cannot be relaxed
until they come out of the nozzle exit considering the process time of 100 μs and the relaxation time of the same order
for most polymers.
Even though the present research has been performed
with polymeric liquids only, the same principle should be
applied to inks from other materials. Especially, the present
result can also be used even for predicting the jetting behavior of the solution of a flexible polymer which can be
regarded as a different class of fluids from the xanthan gum
solution. Considering that suspensions do not show strong
strain hardening, the detachment will be much easier. However, since most inks contain polymers and/or surfactants
for suspension stability, the strain hardening can affect the
detachment to a certain degree. As many different kinds of
polymers and large molecules of various shapes are used
in electronic industries such as organic light-emitting diode
displays and polymer light-emitting diodes, the results of
the present research will be valuable information for those
industries.
Acknowledgement This work was partially supported by Midcareer Researcher Program through NRF grant funded by the Ministry
of Education, Science and Technology, Korea (no. 2010–0015186).
Rheol Acta (2013) 52:313–325
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