Numerical Simulation of Thermal Inkjet Droplet Ejection

Abstract

Numerical simulation permits exploration of a system design
space prior to manufacture. Here, we present and discuss the results
of numerical simulations of the droplet ejection process in
thermal inkjet printers. We compare the simulation results with
those from prior experimental and numerical investigations, and
then consider the performance of novel suspended heater actuators.

Full text

NUMERICAL SIMULATION OF THERMAL INKJET DROPLET EJECTION
Mallinson S.G.*, McBain G.D., Horrocks G.D., Reichl P.J.,
O’Mahony A.P., Pye N.B., Barber T.J. and Yeoh G.H.
∗Author for correspondence
Memjet, Macquarie Park, NSW, 2113, Australia,
E-mail: sam.mallinson@memjet.com
NOMENCLATURE
α[-] Volume fraction
C[-] Interfacial compression coefficient
d[m] Diameter
∆[m] Cell size
φ[kg m−2s−1] Mass flux
F[N m−1] Body force
I[A] Heater current
k[m−1] Interfacial curvature
λ[-] Pressure function parameter
µ[Pa s] Viscosity
ˆ
n[-] Unit normal, positive into liquid
p[Pa] Pressure
R[Ω] Heater resistance
ρ[kg m−3] Density
σ[Pa m] Surface tension
S[m2] Surface area
U[m s−1] Velocity
t[s] Time
T[K] Temperature
Subscripts
b Bubble
e Pressure function parameter
f Cell
g Gas phase
h Heater
`Liquid phase
n Nucleation
s Saturated vapour
∞Ambient
ABSTRACT
Numerical simulation permits exploration of a system design
space prior to manufacture. Here, we present and discuss the re-
sults of numerical simulations of the droplet ejection process in
thermal inkjet printers. We compare the simulation results with
those from prior experimental and numerical investigations, and
then consider the performance of novel suspended heater actua-
tors.
INTRODUCTION
Thermal inkjet actuators work by heating the ink adjacent
to thin electrically conductive films (called heaters) at approx-
imately 1 billion kelvins per second. This pushes the ink past
the normal boiling point until it reaches the superheat limit,
approximately 90% of the critical temperature, at which point
the ink boils explosively, with an initial bubble pressure of ap-
proximately 10 MPa. The subsequent bubble growth forces ink
through a nozzle as a long slug, and this slug eventually forms
one or more ink droplets that then strike the printing media.
Here, we discuss and present numerical simulations of the
bubble growth and droplet ejection process. The simulations
have been performed using the OpenFOAM-6 solver interFoam,
and we have modelled the bubble pressure variation with time as
an exponentially decaying pressure pulse, originally described
by Asai [1], and subsequently used in several other studies of
droplet ejection, for example [2; 3; 4; 5].
First, the numerical method is described, including a detailed
explanation of the model used to simulate bubble growth. This
model requires a number of thermodynamic properties, and the
techniques used to compute them are presented and discussed.
The method is then applied to the original Canon bubble-jet ex-
perimental configuration [1], with good agreement between the
earlier results and the present calculations. Subsequently, the
method is used to simulate the performance of novel suspended
heater thermal inkjet actuators, which are used in the first gener-
ation of Memjet printer systems [6].
NUMERICAL METHOD
Flow equations
In the interFoam solver, the continuity (1) and Navier–
Stokes (2) equations are solved together with a transport equation
for the volume fraction, α(3) [7].
∇·U=0 (1)
ρDU
Dt =−∇p+ρg+∇·µ∇U+∇UT+F(2)
Dα
Dt =0 (3)
Here the surface tension, previously part of the boundary condi-
tion on the interface, is represented by a continuous volumetric
body force [8]:
F=σk∇α (4)

where the interfacial curvature is given by:
k=∇·(∇α/|∇α|)(5)
The fluid properties are calculated as the volume fraction
weighted-average:
ρ=αρ`+ (1−α)ρg(6)
µ=αµ`+ (1−α)µg(7)
where αis one in the liquid phase, zero in the gas phase and takes
intermediate values near the interface.
The interface between the phases is typically smeared over
several cells in the standard VOF method. To provide sharper in-
terfaces, interFoam provides an additional compression term to
the LHS of Equation (3): −∇[α(1−α)Ur]. The artificial com-
pression velocity is given by:
Ur=ˆ
nfminC|φ|
|Sf|,max|φ|
|Sf| (8)
where ˆ
nfis the unit outward vector normal, the max function op-
erates over the entire domain and the min function only operates
locally at each interface cell [9]. The compression coefficient, C,
sets the level of compression; if it is equal to zero, there is no
compression, whilst setting it to one gives a balance between the
desired interface compression and unwanted parasitic currents
which can be generated by the compression [9; 10]. Here, we
have used C= 1.
Geometry and mesh
Two different meshing tools were used in this study. For the
relatively simple geometry of the Canon bubble-jet, the Open-
FOAM blockMesh utility was used, with the cell size for the
baseline simulations matching that of finest used in the original
study, ∆=3µm. A close up view of the mesh is shown in Fig. 1;
note that a symmetry plane was used, running along the centre of
the simulation domain, and gap in the wall is where the heater is
located.
For the more complicated Memjet geometry, the OpenFOAM
snappyHexMesh utility was used. The geometry was generated
by: exporting the MEMS mask edges as a series of points from
GDSII format using KLayout [11]; importing those edges, creat-
ing wires and faces, and then extruding volumes and performing
boolean operations using the FreeCADv0.17 Python API [12];
the generated geometry was saved as a .STEP file, the surface
groups for each of the patches — bottom inlet, top outlet, heater
and walls — identified, and thereafter surface meshes for each
of the patches were created using Gmsh 4.0 [13], and exported
Figure 1. Close-up view of the mesh on the walls for Canon
bubble-jet simulations. Heater coloured red.
to .STL format for use in snappyHexMesh. The walls including
the heater are shown in Fig. 2. A scanning electron micrograph
for the manufactured device is shown in Fig. 3; in this view, the
device has been sectioned to show the interior of the firing cham-
ber, with the heater in the centre, the nozzle at the top, and the
feedhole towards the bottom.
Discretization and solvers
The transient terms are solved using Crank–Nicolson time-
stepping, and the time step is set using a maximum Courant
number of 0.2. The spatial terms are discretized using second-
order centred-differencing. Pressure coupling is achieved using
the PISO method. The pressure equation is solved using the di-
agonal incomplete-Cholesky preconditioned conjugate gradient
method. The velocity and VOF equations are solved using the
symmetric Gauss–Seidel method; the interface compression term
is solved using the MULES technique [15].
Material properties
The original Canon bubble-jet simulations considered a range
of different ink formulations; here we perform simulations for
one particular formulation, called ink A, for which the material
property values are shown in Table 1. For the simulations of
the Memjet device, we assumed the properties of an ink used
in our first generation printers, at a temperature of 35◦C, which
is typical of the local printer working conditions. For this ink,
the viscosity values come from measurements using a spindle
viscometer, the surface tension comes from measurements using
a du No¨
uy ring, the nucleation pressures were estimated using
the generalized corresponding states method [14], and the value

Figure 2. Memjet device simulation domain. Heater coloured
red. The left image shows the interior of the chamber (show-
ing only half the domain), with a feed-hole supplying ink to the
chamber, and a heater which develops a bubble and forces ink
through the nozzle. The right image shows the plan view. For
reference, the nozzle width is 12 µm whilst the heater width is
3 µm.
Figure 3. Scanning electron micrograph of the Memjet device
which has been segmented through the centre (compare with left
image above). Heater coloured white.
of ps(T∞)is calculated as the mole fraction average.
Boundary conditions
At the wall, no-slip conditions are applied for velocity, to-
gether with zero gradient for pressure, which includes a correc-
tion for surface tension terms. The boundary conditions on the
volume fraction are set as a constant contact angle. At the in-
let and outlet boundaries, the pressure is held constant during
outflow, whilst the total pressure is constant during inflow; zero
gradient conditions are applied to the normal velocity, with the
tangential velocity set to zero.
The bubble growth is modelled by setting the liquid volume
Case ρ
(kg m−3)
µ
(mPa s)
σ
(mPa m)
pn
(MPa)
ps
(kPa)
ink A [1] 1000 4.5 53.8 7.5 16
VersaPass ink 1037.3 2.0 39.7 9.8 5.6
Table 1. Summary of liquid properties used in the simulations.
fraction at the heater equal to zero, and applying a transient pres-
sure function, given by Asai [1]:
pb= [pn−ps(T∞)]exp"−t
teλ#+ps(T∞)(9)
For the Canon bubble-jet, Asai [1] gives te=17 µs and ps(T∞) =
16 kPa. For the Memjet simulations, the value of teis calculated
according to the method outlined in [16]. A thorough discussion
of this equation was given earlier [17]. Asai showed that the bub-
ble pressure at times just after nucleation could be determined
from the rate of change of enthalpy with time. By equating the
pressure at the end of the early stage with the assumed pressure
variation in the later stages, Eqn. (9), it was shown that (see [16],
Eqn. (28)):
te=t31+t3
t1−1/λt3
t2−1/(2λ)
(10)
where t1,t2and t3are functions of liquid material properties, heat
flux into the liquid, heater area and heater driving point inertance;
they are given in [16], Eqns. (23) and (25).
Figure 4. Plot of temperatures above the boiling point during
voltage pulse, together with nucleation probability.
The inertance is calculated here using solutions of the Laplace
equation, as described in [18]. The heat flux into the liquid at the
point of nucleation is determined from one-dimensional finite-
element calculations using constant material property values, and
with a constant volumetric load provided by Joule heating, I2R.
The finite element solution proceeds until the volumetric bub-
ble embryo nucleation rate is such that the nucleation probabil-
ity [16], also called the bubble reliability [19], is greater than 0.5.
Figure 4 shows the temperature in the thermal boundary layer on

the heater, for temperatures above 100◦C, along with the nucle-
ation probability. We can see that in this instance, nucleation
occurs approximately 350 ns after heater switch on. Using the
value of heat flux to the liquid at this instant, we find that for the
Memjet device, te=0.017 µs for λ=0.5.
The adopted pressure inlet boundary condition can result in
liquid escaping the system via the heater, which is clearly un-
physical. This can be resolved by monitoring the flux through
the heater and switching to an impermeable wall condition as
soon as any liquid attempts to exit.
RESULTS: CANON BUBBLE-JET
The results from the present numerical simulations of the
Canon bubble-jet are compared with the experimental and nu-
merical data for volume fraction in Fig. 5. The comparison be-
tween the present and earlier results is good. A more quantitative
way to compare the results is via the droplet volume and velocity.
Figure 6 shows the current predictions for the transient variation
of average velocity and ejected along with the simulated values
reported in [1]. The earlier values are reasonably close to the
asymptotic levels predicted here.
Figure 5. Comparison of Asai CFD (left) and experiment (cen-
tre) with current CFD (right) for time–sequence = 2, 7, 12, 17,
22, 27, 32 and 37 µs (top to bottom)
Figure 6. Comparison of predicted volume-weighted velocity
and ejected volume with those from Asai [1]
RESULTS: MEMJET VERSAPASS
Asai [1] used λ=0.5 for simulations of the Canon bubble-jet,
but then noted [16] that the value of λshould be determined by
comparison with experimental data. We obtained droplet veloc-
ity values from sequences of high resolution stills using a strobo-
scopic imaging system. Droplet volume was estimated by firing a
large number droplets into a beaker placed on a balance, and then
dividing the mass increase by the number of droplets ejected,
with account taken of the ink evaporation from the beaker dur-
ing the test by measuring the rate of mass loss from the beaker
during inactive periods.
For the simulations of the Memjet device, we first used λ=
0.5; the comparison with experimental data is poor, see Fig. 7.
We then proceeded to explore higher values of λ, finding that
for λ=0.575, the velocity is slightly less than the experimental
value, whereas the volume is slightly higher. Runs with one and
two levels of mesh adaption for this value of λshow only minor
differences to the original mesh results.
Figure 7. Comparison of predicted volume-weighted velocity
and ejected volume for the Memjet device with different values
of λ.
Figure 8 presents a sequence of time frames showing the
predicted free-surface motion for the Memjet device using λ=
0.575, for which te=0.013 µs. A slug of liquid emerges from
the nozzle, and very soon thereafter, the slug begins to narrow
near the nozzle, forming an extended tail. The tail pinches off
near the main droplet just before 4 µs. The part of the tail which
is still attached to the main ink body then breaks away just before
5 µs. This results in the formation of three small satellite droplets
after 5 µs, two of which subsequently merge.
If the bubble grows to fill the region between the heater sur-
face and the nozzle exit plane, as occurs with the Memjet device,
it can cause the liquid film to tear and the vapour bubble to vent
to atmosphere. Thereafter, part of the torn meniscus can move
back into the chamber and strike the heater. This is shown in
Figs. 9 and 10. During the early stages, the bubble grows and
forces ink through the nozzle. By 0.6 µs, the ink near the nozzle

Figure 8. Predicted meniscus location. Upper frame: early
times; lower frame: later times.
rim begins to move downwards, with the central part of the stalk
reaching approximately −50 m s−1. Very shortly thereafter, the
boundary condition switches, and the velocity of the ink on the
heater reduces. If the bubble is relatively smaller, such as in the
Canon bubble-jet, venting does not occur and instead the bubble
collapses which can cause pitting of the heater [21; 22].
To explore the effect of the thickness of the liquid layer be-
tween the heater and chamber roof, simulations were performed
for geometries with this thickness equal to two- and four-times
the original value. A comparison in presented in Fig. 11 of the
free surface location at the switch-over point, which is different
for each geometry, and at the end of the simulation run time. It
can be seen that doubling the thickness does not prevent venting,
but by quadrupling the value, the bubble does not seem to vent.
It is also seen that increasing the liquid layer thicknes causes a
much longer droplet tail, which will inevitably break into several
smaller satellites. For the larger thickness values, the tail seems
to leave behind a small puddle on top of the roof layer, which
could cause issues with droplet misdirection.
CONCLUSION
Numerical simulations of the operation of two different ther-
mal inkjet actuators have been presented and discussed. The
comparison between the original Canon bubble-jet simulations
and experiment using the current method is good. For the Mem-
jet device, tuning the pressure boundary condition was required
Figure 9. Side view with cut through nozzle centre, show-
ing isosurface of volume fraction = 0.5, coloured by z-velocity.
Heater coloured red.
Figure 10. Tilted view, showing isosurface of volume fraction
= 0.5, coloured by z-velocity. Heater coloured red.
to obtain good agreement with experiment. It was also shown
that the large bubble produced by the Memjet actuator causes
the liquid film to tear and the bubble to vent to atmosphere, thus
avoiding bubble collapse, which is known to cause heater dam-
age.
ACKNOWLEDGMENT
We thank Dr Darrin Stephens of Applied CCM for expert ad-
vice in programming OpenFOAM. This research project was un-
dertaken with the assistance of resources and services from the
National Computational Infrastructure (NCI), which is supported
by the Australian Government.
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