Modeling the Drop Formation Process in Inkjet Printheads
The content for this article was contributed by Herman Wijshoff of Océ Technologies B.V
Inkjet developments in document printing are moving towards higher productivity and quality, requiring adjustable small droplet sizes to be fired at high repetition rates. New industrial applications are also increasing the demand for higher repetition rates as well as for differing droplet properties and droplet materials. Understanding the drop formation processes and all relevant phenomena is the basis for the inkjet technology development at Océ Technologies B.V. Modeling the flow characteristics of these processes with FLOW-3D was a key part of Oce's efforts.
Droplets are measured by means of various optical methods like stroboscopic illumination at drop formation rate and high-speed camera recordings. With laser-Doppler measurements, meniscus movements without drop formation are recorded.
The detailed knowledge behind the drop formation processes enabled the development of drop size modulation techniques , which are of great importance for market applications where Océ is an important player.
These measurements give details on ink flow outside the printhead. Phenomena inside the channels are difficult to measure. Only by using the piezo actuator also as a sensor do we get a recording of the average pressure inside the ink channel, which enables monitoring of jetting stability  and feed-forward control of the driving waveform . More information on the phenomena preceding the drop formation is needed for a better understanding of the physical processes.
Details on ink flow and acoustic pressure waves are only available through modeling . Two main aspects in modeling piezo inkjets are the acousto-elastic interaction and free-surface flow. To model printhead structured dynamics and the interaction with ink acoustics the finite element code Ansys® is used. For free-surface flow with surface tension (drop formation) and its impact on channel acoustics we use FLOW-3D.
Firing of a Droplet
Figure 1: Actuation taking into account channel acoustics.
An electric voltage on a piezo element enlarges the channel
and a negative pressure is generated. After reflection at
the resevoir this pressure is amplified by the second slope
of the driving waveform to get a large positive pressure peak
at the nozzle, which fires a drop.
A long ink channel with a nozzle at the right and a large reservoir at the left is the simplified geometry of the inkjet device as shown in Figure 1. A piezo actuator element drives each channel. To fire a droplet, an electric voltage is applied and the channel cross-section will be deformed by the inverse piezo-electric effect. This results in pressure waves inside the channel. The pressure waves propagate in both directions and will be reflected at changes in characteristic impedance (variations in cross-section and compliance of the channel structure). The first slope of the driving waveform enlarges the channel cross-section, and the resulting negative pressure wave will be reflected at the reservoir at the left. The reservoir acts as an open end and the acoustic wave returns as a positive pressure wave. The second slope of the driving waveform removes the driving voltage. This will reduce the channel cross-section to its original size and will amplify the positive pressure wave when tuned to the travel time of this acoustic wave.
To model the acoustics with FLOW-3D, a user routine was written to include wall flexibility. One wall is a rigid obstacle and becomes the (piezo) actuator by applying a movement according to the driving waveform and the resulting piezo deformation as calculated with Ansys®. Another wall is divided into many segments, each moving towards a new position corresponding to the compliance of the channel cross-section, which also is calculated with Ansys®. A damping parameter was used in this routine, not only for numerical stability but also to include a first order effect of printhead dynamics.
Figure 2: Acoustic model in FLOW-3D. Actuation is done with a moving rigid obstacle and wall flexibility is incorporated with pressure dependent movement of another wall. The wall is divided into many segments to enable a response on local pressure variations.
Figure 3. Pressure at the nozzle entrance. First a negative
pressure retracts the meniscus surface, the fill-before-fire
action. Then a large postitive pressure peak reaches the nozzle,
which drives the ink through the nozzle. In the small cross section
of the nozzle, acceleration of the ink results in drop formation.
The main effect of this acousto-elastic interaction is the reduction of the effective speed of sound from typical 1200 m/s to 800 m/s. The resulting pressure at the nozzle entrance is shown in Figure 3, compared with an acoustic model written in Matlab® without mean ink flow and drop formation. The main effect of the drop formation process on acoustics is that the positive pressure peak will be reduced, because of the variable amount of ink in the nozzle. This reduces the efficiency and explains the quantitative difference between acoustic models in Matlab® or Ansys® and experiments.
A relatively large grid size of several micrometers is used for the acoustic calculations. For details of the drop formation itself, a very fine grid is needed. The calculations are therefore split into a 3D acoustic part with a coarse grid and a 2D part with cylindrical coordinates and a fine grid, with the pressure at the entrance of the nozzle input as a boundary condition taken from the 3D computation. A simulation of the drop formation at a very high drop speed is shown in the animation.
Figure 4: Measured and calculated drop formation at a very high drop speed.
A supercritical acceleration of the first amount of ink results in a fast
satellite drop, because surface tension forces are no longer capable of holding the ink together.
Even the finest details of tail break-up are predicted very well, as shown in Figure 5 where the formation of a secondary tail is a consequence of the tail going through different flow regimes before it breaks off. The detailed knowledge behind the drop formation processes enabled the development of drop size modulation techniques , which are of great importance for market applications where Océ is an important player.
Figure 5: Measured and calculated formation of a secondary tail before tail break-up. The flow regime goes from a viscous flow to a kind of inviscid flow when tail width becomes smaller than the viscous length scale. The radius of the secondary tail is only 1 micrometer.
In order to fire drops at very high repetition rates, free-surface flow in the nozzle and the acoustics in the channel are designed to give a very strong refill of the nozzle. The refill mechanism is a part of the FLOW-3D acoustic simulations and is also driven by the variable amount of ink in the nozzle during the drop formation cycle. However, a strong refill mechanism can lead to overfill at low frequencies . As a result, in some cases wetting of the nozzle-plate occurs. With wetting, all kinds of flow phenomena on the nozzle-plate are visible  and FLOW-3D simulations help to identify the disturbing mechanisms that can result from wetting. Drop properties will change and air may be entrapped  as shown in Figure 6. Air bubbles are the main cause of instabilities in the drop formation process and understanding the dynamic behaviour of air bubbles in the flow and acoustic field near a nozzle is of crucial importance to realise jetting stability at high repetition rates.
Figure 6: Air entrapment from a wetting layer, the onset of a possible disturbance of the drop formation process, and a net displacement pattern after one drop formation cycle of a 10 µm air bubble as function of its position on the axis of a nozzle.
Figure 7: Impact, kinetic spreading and cooling
of an ink drop at 130° C on a cold (room temperature)
surface after 0, 2, 5, 10, 50 and 500 µs.
The final stage in the inkjet printing process is the impact and spreading of a droplet on the surface of some kind of medium. Depending on the kind of ink, the spreading of the droplets stops by some kind of drying process: evaporation with water and solvent based inks, solidification with amorphous phase change inks, crystallisation with crystalline phase change inks, polymerisation with UV-curable inks, etc. As an example, the impact and kinetic spreading of a hotmelt ink drop is shown in Figure 7.
Understanding the drop formation process is the first step to realize maximum jetting stability and to control and manipulate the drop formation and impact behavior. With our modeling we can predict the resulting drop properties very accurately, and we can identify potential new inkjet devices by exploring new actuation principles, new acoustic principles and new drop formation mechanisms. Modeling with FLOW-3D is a key part of the whole sequence of simulations. Future developments will move more and more towards the generation of smaller and larger drops and will go to higher drop-on-demand frequencies. We will also explore the behavior of new materials, for example the jetting of pure metals is a possible application.
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 M.B.Groot Wassink, “Inkjet printhead performance enhancement by feedforward input design based on two-port modelling,” Ph.D. thesis, Delft (2007)
 H. Wijshoff, “Free surface flow and acousto-elastic interaction in piezo inkjet,” Proc. Nanotech2004 2, 215 (2004)
 H. Wijshoff, “Manipulation drop formation in piezo acoustic inkjet,” Proc. IS&T’s NIP22, 79 (2006)
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