International Journal of Mechanical Engineering and Technology (IJMET)
Volume 10, Issue 04, April 2019, pp. 430–442, Article ID: IJMET_10_04_042
Available online at http://www.iaeme.com/ijmet/issues.asp?JType=IJMET&VType=10&IType=4
ISSN Print: 0976-6340 and ISSN Online: 0976-6359
© IAEME Publication
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A NUMERICAL SIMULATION AND
VALIDATION STUDY OF THE
MATHEMATICAL MODEL OF DROPLET
FORMATION IN DROP ON DEMAND INKJET
PRINTER AND THE EFFECT OF
RHEOLOGICAL PROPERTIES OF
POLYMERINK FOR AUTOMOBILE LIGHTING
APPLICATION
Rajesh.P.K., and Aravindraj.S
Department of Automobile Engineering, PSG College of Technology,
Coimbatore, Tamilnadu. India.
ABSTRACT
A majority of the modern inkjet printers utilise drop on demand devices because of
its precision in terms of time and easy control. The time-dependent fluid interface
disruption renders the fluid dynamics process during droplet ejection complex. The
current work attempts to provide an idea of the drop ejection behaviour based on the
computation of energies required for droplet formation and splat formation. The
simulation results for various nozzle diameters with different polymer inks are
examined and it is validated with computational model. Further attempt is made to
analyse the effect of rheological properties like viscosity and surface tension in the
droplet formation.
Key words: Drop on demand; inkjet; droplet ejection; viscosity; surface tension.
Cite this Article: Rajesh.P.K. and Aravindraj.S, A Numerical Simulation and
Validation Study of the Mathematical Model of Droplet Formation in Drop on
Demand Inkjet Printer and the Effect of Rheological Properties of Polymerink for
Automobile Lighting Application, International Journal of Mechanical Engineering
and Technology 10(4), 2019, pp. 430–442.
http://www.iaeme.com/IJMET/issues.asp?JType=IJMET&VType=10&IType=4
1. INTRODUCTION
There is a tremendous efforts in the search for new means of further improving the quality
and reducing device cost of ink jet printing due to its rapid growth over manufacturing of
automotive electronics. Knowledge in fluid dynamic process of drop formation and drop
ejection takes precedence in research and development of new ink jet print heads. There are
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Rajesh.P.K. and Aravindraj.S
two main types of ink jet devices, namely, continuous–jet type and drop-on demand type. [3 4]. In a continuous-jet device, there is a disintegration into a train of drops of the liquid,
emerging from the nozzle continually in the form of a jet. The amount of electric charge on
each individual drop and direction of motion of each drop from the continuous jet require
sophisticated electrical signals. A drop-on-demand device, on the other hand, uses electrical
signal to control the actuation at the instance of the ejection of an individual drop. Due to its
basic simplicity, the drop-on-demand type is common in the most modern ink jet printers.
This work focuses on the basic drop ejection process in drop-on-demand devices.
A drop-on-demand inkjet printer consists of a fluid chamber with a nozzle which is
actuated to eject the droplet. The actuation pushes a certain amount of the liquid out of the
fluid chamber through the nozzle. A drop is ejected when the liquid pushed out of the nozzle
gains enough forward momentum to overcome the surface tension restoring effect, [5]. The
generation and behaviour of liquid droplets [6] is effected by the Surface tension, inertia and
viscosity. Surface tension is a contractive tendency of the surface of a liquid that allows it to
resist an external force. Liquid atoms or molecules at a free surface have higher energy than
those inside the liquid body. Therefore, the shape of liquid with the lowest surface tension
energy is sphere. For the generation of a droplet, a liquid must necessarily have the tendency
to form a shape with lowest energy. The attraction of water molecules to each other is greater
than the molecules in the air, when the droplet is generated. As a result, an inward force at its
surface makes water to behave as if its surface was covered by a stretched elastic membrane.
This is also the primary cause of pinch‐off effect [7]. The surface tension of most of the
liquids used in inkjet printing have the order of tens of dyn/cm (or mN/m). The importance
and influence of the above parameters can characterized by three essential dimensionless
numbers: 1. Reynolds number, 2.Weber number and 3.Ohnesorge number [8].
Figure 1 Stages of droplet formation process [9]
The ink inside the nozzle stays at equilibrium state, before the nozzle gets actuated. Ink
velocity and pressure are zero at initial stage (A). A high pressure is generated inside the
nozzle when the nozzle gets actuated, and the liquid start to flow out from the nozzle orifice
(B). Kinetic energy is transported from the actuator walls to outflow and it undergoes an
attenuation process, in order to overcome the resistance from surface tension (C). The droplet
is then connected with the liquid inside the nozzle by a skinny fluid filament (D). When the
liquid column momentum is large enough, the droplet will escape from the nozzle (E).
Surface tension acts as a force to pinch off the ligament. The meniscus retracts inside the
nozzle (F).
In the inkjet printing applications, a single droplet is invariably desired but due to surface
tension additional satellite droplets which are usually smaller than intended primary droplet,
are formed and they cause several problems in printing [10]. When the unexpected satellite
droplets land on places other than where primary droplets do, they result in the degradation of
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A Numerical Simulation and Validation Study of the Mathematical Model of Droplet Formation in
Drop on Demand Inkjet Printer and the Effect of Rheological Properties of Polymerink for
Automobile Lighting Application
print quality, which further leads amination or failure. This is most clearly indicated by the
blurring of the trailing edge of a printed area [11].
The drop formation phenomenon is theoretically, governed by Navier–Stokes equations
with appropriate boundary conditions describing fluid interface motions [12]. The
conventional methods cannot be used to obtain the desired mathematical solutions because of
the prevalence of the nonlinearities arising from inertia, capillarity, and coupling of the free
surface kinematics to the flow field. Hence, a Computational Fluid Dynamics (CFD) package
FLOW-3D V 10.0 [13] is utilised to simulate the complex fluid dynamic process during drop
ejection and it is validated with energy model.
Some other researches on droplet formation process in Drop-On-Demand inkjet printing
need to be analysed to overcome the problem. A quantum of research work has been carried
out in the mechanism of droplet formation, especially on Newtonian fluids, subsequent to the
invention of the first DOD inkjet printer in the 1940s. In the recent years, as more Non‐
Newtonian fluids have been widely used in manufacturing of automobile electronics and a
bulk of research on the droplet formation of Non- Newtonian fluids are carried effectively.
Due to the tens of micron length and the time scale of less than a hundred micro‐second,
micro scale droplet generation differs from macro scale droplet generation. Shin et al. [14]
and Verkouteren et al. [15] analysed the transient process of droplet generation using a charge
coupled device (CCD). Dong [16] and Carr [17] studied the dynamics of drop‐on‐demand
(DOD) droplet formation using an imaging system with an inter‐frame time of 1 μs. The
experiment was conducted on a viscosity range of (1.0 ‐ 5.0 cP) and surface tensions (35 – 73
mN/m). They investigated the stages of droplet formation including the ejection and
stretching of liquid and the pinch‐off of liquid thread from the nozzle exit. Lopez et al. [18]
studied the combination effect of ink rheological behaviour focusing on the dynamics of
filament break‐up and effect of rheological properties on droplet formation. Cittadino et al.
[19] developed a non‐linear model to predict the velocity of the ejected droplet, based on a
balance of forces, showing that the ejection velocity is a strong function of the applied
voltage. Feng and James [20] proposed an comparatively simpler approach based on a series
of numerical calculations on Flow3D. This reference has given an idea about the droplet
ejection behaviour for establishing nozzle head design and shows that the volume of ejected
droplet is very close to the volume of fluid pushed through the nozzle by an actuation pulse.
2. NUMERICAL SIMULATION FOR DROPLET FORMATION BASED
ON ENERGY APPROACH
The energy required to form the droplet are equated in energy approach, to find the diameter
of the droplet. Based on the law of conservation of momentum, the energies before and after
impact are equated to find the splat diameter (i.e.) Diameter of the spread after fall on the
substrate [21].
2.1. Energy Required to Eject Single Droplet from the Nozzle
These energies are required to eject the droplet is as follows:

Energy [E1] required to deflecting the membrane.

Frictional energy [E2] in the orifice.

Kinetic energy [E3] of droplet at the outlet of print head.

Surface tension energy [E4] of the droplet at outlet.
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The total energy [E] required to eject the droplet from nozzle should be greater or equal to
the sum of these energies.
E=E1+ E2+ E3+ E4
2.2. Volume of the Droplet Eject from the Nozzle
The volume of the droplet ejected from the nozzle is equal to the maximum volume displaced
by the deflecting membrane. The driving voltage on the piezoelectric device is converted into
required force [F] acting on the centre of the disc.
The maximum deflection in the disc is given by the formula [22]:
(1)
The maximum deflection is obtained by substituting
r=0, = 0.3 and I
, we get
(2)
By integrating the volume displaced in small layer we get the total volume of the ink
displaced (i.e.)
Total volume of the ink displaced is given by
∫
(3)
On integrating Equation 3, we get
(4)
On simplifying the equation 4, we get
(5)
2.3. Diameter of the Droplet Formed from the Nozzle
The diameter of droplet is calculated by equating the volume of the sphere and the volume of
the ink displaced (i.e.)
(6)
By simplifying the equation 6, we get
√
(7)
Equation 7 gives the diameter of the spread which depends on the ink chamber diameter
and the maximum defelection of the membrane.
2.4. Energy Required for Deflecting the Membrane [E1]:
The energy required for deflecting the membrane is equated to the product of force [F] and
the maximum deflection (i.e.)
(8)
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A Numerical Simulation and Validation Study of the Mathematical Model of Droplet Formation in
Drop on Demand Inkjet Printer and the Effect of Rheological Properties of Polymerink for
Automobile Lighting Application
2.5. Frictional Energy Required at the Orifice [E2]:
Friction loss (or skin friction) is the loss of pressure or “head” that occurs in nozzle flow due
to the effect of the fluid's viscosity near the surface of the orifice [23]. The fictional loss is
given by the Hagen-poissoulle equation (i.e.)
(9)
Frictional energy[E2] is given by
E2 =
(10)
2.6. Kinetic Energy of Droplet at the Outlet of Print Head [E3]:
When the droplet moves with velocity ub, it possess kinetic energy [E3]
(11)
Substituting the value of V from Equation 5 to Equation 11, we get
[
]
(12)
2.7. Surface Tension Energy Of Droplet At The Outlet [E4]:
The surface tension of the liquid is given by
(13)
The surface tension energy is given by
(14)
2.8. Total Energy Required for Eject the Droplet [E]:
E=E1+ E2+ E3+ E4
E=
+
+
+
(15)
Equation 15 refers to the total energy required to eject the droplet from the nozzle. The
energy must be greater or equal to the sum of all four energies in order to actuate the nozzle to
eject the droplet.
2.9. Tail and Pinch off Velocity
The pinch-off time coincides with the zero crossing of the ejection velocity (slender-jet
approximation) at the instance of the droplet ejection. In the slender-jet approximation with
neglected radial momentum, the stretching rate tends to become infinity at the nozzle when
the ejection velocity decreases through zero [24]. An instantaneous pinch off is indicated by
an infinite stretching rate. This pinch-off is, therefore, imposed through the boundary
condition and the approximations in the mathematical model. The imbalance in the capillary
tension at the end of the tail causes the formation of the tail droplet when the tail pinches off
at the meniscus. The capillary tension pulls the tail droplet toward the head droplet [25-26].
Simultaneously, the tail droplet mass increases as it sweeps up the ink in the tail, slowing
down the droplet. As a consequential combined effect, the velocity of the tail droplet relative
to the ink in the tail is independent of both viscosity and the size of the tail droplet. To
calculate this velocity, this problem is to be considered in a frame of reference in which the
tail droplet is stationary.
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The forces on the control volume are advection
, Laplace pressure
, and
the direct effect of surface tension
, should sum up to zero if the velocity is constant.
(16)
From this equation, we get the tail droplet velocity as
√
3. DIAMETER OF THE DROPLET AFTER SPREAD [D1]
The diameter of the droplet after spread is calculated by equating the energy attained by the
droplet before impact to the energy attained by the droplet after impact (law of conservation
of energy)
E3+E4+Potential energy = E7+E8
The droplet attains the surface tension energy and kinetic energy before the impact and
due to the pressure applied at the membrane, it possess potential energy [4].
+
+
(17)
The surface tension energy attained when the droplet fall on the substrate is given by [27]
(18)
The energy needed to to spread the droplet in the substrate against the viscososity is given
by [27]
(19)
√
Equating the equation 16,17 and 18, we get
[
]
[
]
√
From equation 19, we infer that the diameter of the spread (d1) and the maximum diameter
of spread is related to the velocity of ejected droplet and the deflection of the membrane,
which is in turn, related to F.
Figure 2 Splat formation on substrate [27]
4. INK AND ITS PROPERTIES
Thermal and electrochemical stability and its high transparency [28-30] make PEDOT: PSS
and PEGDA a conductive material with certain versatility, such as the possibility of
deposition on different substrates. The inks were purchased from Sigma Aldrich, USA. The
Properties of the ink are listed in the Table 1.
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A Numerical Simulation and Validation Study of the Mathematical Model of Droplet Formation in
Drop on Demand Inkjet Printer and the Effect of Rheological Properties of Polymerink for
Automobile Lighting Application
Table 1 Properties of the PEDOT: PSS and PEGDA inks
Ink Properties
PEDOT:PSS
PEGDA
Density
1011 Kg/m3
1120 Kg/m3
Surface Tension
33 mN/m
35.09 mN/m
Viscosity
0.02 kg/ms
0.025 kg/ms
5. SIMULATION OF DROPLET FORMATION FOR INKJET
PRINTING FOR BIO INK
The fluid dynamic analysis of bio-ink droplet formation was done using the Computational
Fluid Dynamics (CFD) package FLOW-3D software, developed by Flow Sciences Inc., Los
Alamos, New Mexico and widely used for inkjet analysis [31]. Inkjet module in FLOW-3D
facilitates the characterization of the droplet formation of polymer-inks for inkjet printing.
Special features in FLOW-3D include,

Powerful physical modelling capabilities.

Easy meshing with multiple structured blocks.

Ability to refine a mesh, independent of the geometry and based on the required spatial
accuracy.

Ability to solve the typical problems of incompressible laminar viscous flow in bio-printing.

Use of special numerical methods to locate free surfaces and applying the proper dynamic
boundary conditions at those surfaces.
5.1. Inverse Ohnesorge Number
Inverse Ohnesorge number (Z) is a dimensionless number that gives the relationship between
the rheological properties- density, viscosity, and surface tension [32-33].
In inkjet printing, liquids with inverse Ohnesorge Number value in between 1 to 10 are
jettable.
√
(20)
Where is the density of the ink, the surface tension of the ink, the nozzle radius and μ
the viscosity of the ink.
The inverse ohnesorge number for different nozzle diameter for polymer inks is calculated
in Table [2, 3].
Table 2 Ohnesorge Number for PEDOT: PSS
Diameter (
)
Value
20
0.913
25
1.02
30
1.29
35
1.208
40
1.118
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Table 3 Ohnesorge Number for PEGDA
Diameter (
)
Value
20
0.7929
25
0.8865
30
0.9711
35
1.049
40
1.122
5.2. Geometrical Model for Bio Polymer Inks in Flow 3D Software
A model with a cylindrical fluid chamber, movable piston and a nozzle (Figure 3) is created in
FLOW 3D. The surface tension model was coupled with General Moving Objects (GMO)
model for the simulation of droplet formation. The Surface Tension model is activated and
wall adhesion is also activated simultaneously. In order to minimize the wet ability of bioinks the contact angle is specified as 90°, since most of the fluids exhibit adhesion behaviour
at solid surfaces. In the present study, the flow of polymer-inks is simulated using
incompressible, laminar and viscous flow mode. Since the Rheological properties of polymer
inks have great influence on droplet formation behaviour, size and tail length they are given
as inputs, [34].
Figure 3 Nozzle Geometry
6. RESULTS AND DISCUSSION
The simulation results for selected nozzle diameter for two different polymer inks based on
inverse ohnesorge number are discussed and the effect of rheological properties on droplet
formation is investigated.
6.1. Printability of Bio Polymer Inks
Printability of bio-ink is determined by its ability to eject stable and repeatable droplets from
the nozzle [35]. Drop formation can be characterized by a dimensionless quantity known as
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A Numerical Simulation and Validation Study of the Mathematical Model of Droplet Formation in
Drop on Demand Inkjet Printer and the Effect of Rheological Properties of Polymerink for
Automobile Lighting Application
inverse ohnesorge number (Z). Generally, 10>Z>1 is the suitable range for stable drop
generation. For Z<1, viscous dissipation prevents drop ejection. For Z>10, the ink is freely
ejected from the nozzle without significant viscous dissipation. The kinetic energy of the
drop, however, increases leading to rupture of filament and formation of satellites [36]. A
typical inkjet printer nozzle diameter ranges from 20 μm to 50 μm . For PEDOT: PSS and
PEGDA, the suitable nozzle diameter is selected based on the inverse ohnerorge number (Z)
obtained from the Table [5.1, 5.2] as shown in Table [4]
Table 4 Suitable Diameter of Inkjet Printer for polymer inks
PEDOT:PSS
PEGDA
30
40
Suitable Diameter
6.2. Simulation of Droplet Formation of Inkjet Printing for Polymer Inks
Ink jetting was simulated by applying velocity on the piston kept over the ink chamber and
caused by the piezoelectric actuation, causing ink ejection out of the nozzle. The operating
conditions of the piezoelectric actuator such as driving voltage, pulse width, and waveform
affect the droplet formation significantly[37-38].
t=0
t=18
t=37
t=56
t=76
t=114
t=133
t=130
t=152
t=171
Figure 4 Droplet formation of PEDOT: PSS ink for the nozzle diameter 40μm
In figure 4, at frame 1, the valve is open and pressure pulse is active but the total energy is
not enough to form the droplet. At 15 , in frame 2 the valve is closed, the negative pressure
is created on top of liquid inside reservoir which ejects the droplet from the nozzle [39]. At
this stage, the ink neck is sufficiently thin to be broken while the rest is sucked into reservoir.
At 30 , the detached volume of liquid outside the nozzle forms a spherical shape. At
30
tail formation takes place and pinch off occurs at 115
The diameter of the
PEDOT: PSS ink observed from the simulated results is 1.255 cm.
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t=0
t=19
t=38
t=57
t=76
t=95
t=114
t=133
t=152
t=190
Figure 5 Droplet formation of PEGDA ink for the nozzle diameter 40μm
In figure 5, at frame 1, the valve is open and pressure pulse was active but total energy is
not enough to form the droplet. At 19 , in frame 2 the valve is closed,a negative pressure is
created on top of the liquid inside the reservoir which ejects the droplet from the nozzle [40].
At this stage, the ink neck is sufficiently thin to be broken while the rest is sucked into the
reservoir. At 57 , the detached volume of liquid outside the nozzle forms a spherical shape
and pinch off occurs at 152 and when compared to PEDOT:PSS ink, the pinch off time in
PEGDA ink is more due to high viscosity.
6.3. Comparison of Numerical Approach With Simulated Results:
From the simulated results, the diameter from droplet is observed to be 1.255 cm and in
numerical approach, by substituting the values [Table 5] in the equation 7, the diameter of the
droplet is observed to be 1.23 cm.
Table 5 Parameters for Force profile used in printing PEDOT: PSS [41]
Pulse Profile Parameters
Force Required to Deflect the Membrane
Rise Time (R)
Fall Time
Thickness of the Membrane
Setting
55 N
30 s
20 s
0.01 m
7. CONCLUSIONS
In the present work, using polymer inks and simulating different nozzle diameters with
FLOW-3D, provided considerable fundamental insights into the factors that control the
performance of a drop-on-demand inkjet printer. Energy equations were developed to
simulate the droplet formation in DOD inkjet printer. These were used to find the droplet
diameter and diameter of the spread as well as the pinch off velocity of the droplet. From the
results given here, properties of the flow that may be useful in the development and design of
suitable print head in the laboratory for experimental purpose, can be suggested.
From the simulated results, it is evident that the efficiency of droplet generation from drop
on demand print‐head depends on the viscosity, surface tension, nozzle size, density, and the
driving waveform like wave shape, frequency, and amplitude. The comparison of the results
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A Numerical Simulation and Validation Study of the Mathematical Model of Droplet Formation in
Drop on Demand Inkjet Printer and the Effect of Rheological Properties of Polymerink for
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obtained from the numerical approach and the simulated results shows that the numerical
approach values are slightly deviated from the simulated value.
NOMENCLATURE
S No
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
Symbols
E1
E2
E3
E4
E
r
D
E
I
t
d
uo
do
hf
V
ub
p
d1
Re
Vi
Description
Energy required for deflecting diaphragm
Frictional energy in orifice
Kinetic energy of the droplet at the outlet of the print head
Surface tension energy of the droplet at the outlet
Total Energy to eject the droplet
Local radius of the membrane from its centre
Diameter of the chamber
Young's modulus of the membrane
Flexural rigidity of the membrane
Poisson's ratio of the membrane (assumed as 0.3)
Thickness of the membrane
Maximum deflection of the membrane
Diameter of the droplet
Apparent viscosity
Average velocity of the fluid inside orifice
Diameter of the orifice
Density of the ink
Frictional head
Volume of the droplet
Velocity of the droplet at the beginning of its ejection
Pressure inside the droplet
Contact angle
Diameter of the spread
Reynold’s number
Droplet impact velocity
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