The dynamics of the piezo inkjet printhead operation (PDF

The dynamics of the piezo inkjet printhead operation (PDF
Physics Reports 491 (2010) 77–177
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Physics Reports
journal homepage: www.elsevier.com/locate/physrep
The dynamics of the piezo inkjet printhead operationI
Herman Wijshoff
Océ Technologies B.V., P.O. Box 101, 5900 MA, Venlo, The Netherlands
article
info
Article history:
Accepted 11 March 2010
Available online 31 March 2010
editor: I. Procaccia
Keywords:
Inkjet
Piezoelectricity
Acoustics
Drop formation
Bubble dynamics
Wetting
abstract
The operation of a piezo inkjet printhead involves a chain of processes in many physical
domains at different length scales. The final goal is the formation of droplets of all kinds
of fluids with any desired volume, velocity, and a reliability as high as possible. The
physics behind the chain of processes comprise the two-way coupling from the electrical
to the mechanical domain through the piezoelectric actuator, where an electrical signal is
transformed into a mechanical deformation of the printhead structure. The next two steps
are the coupling to the acoustic domain inside the ink channels, and the coupling to the fluid
dynamic domain, i.e. the drop formation process. The dynamics of the printhead structure
are coupled via the acoustics to the drop formation process in the nozzle. Furthermore,
wetting of the nozzle plate and air bubbles can have a negative influence on the printhead
performance. The five topics (actuation, channel acoustics, drop formation, wetting, and air
bubbles) are reviewed in this paper. This research connects the product developments for
many emerging new industrial applications of the inkjet technology to the fundamental
physical phenomena underlying the printhead operation.
© 2010 Elsevier B.V. All rights reserved.
Contents
1.
2.
Introduction............................................................................................................................................................................................. 79
1.1.
Historical overview on inkjet technology ................................................................................................................................. 79
1.2.
Printing principles ...................................................................................................................................................................... 85
1.3.
Printhead operation.................................................................................................................................................................... 86
1.3.1.
Working principle........................................................................................................................................................ 86
1.3.2.
Printhead testing ......................................................................................................................................................... 88
1.4.
Guide through the sections ........................................................................................................................................................ 90
Structure dynamics ................................................................................................................................................................................. 90
2.1.
Piezoelectricity............................................................................................................................................................................ 90
2.2.
Actuating ..................................................................................................................................................................................... 93
2.2.1.
Bump mode actuation ................................................................................................................................................. 93
2.2.2.
Bend mode actuation................................................................................................................................................... 95
2.2.3.
Shear mode actuation.................................................................................................................................................. 96
2.3.
Actuation efficiency .................................................................................................................................................................... 97
2.4.
Local cross-talk ........................................................................................................................................................................... 98
2.4.1.
Direct cross-talk........................................................................................................................................................... 98
2.4.2.
Pressure-induced cross-talk........................................................................................................................................ 99
2.5.
Printhead dynamics .................................................................................................................................................................... 100
2.5.1.
Modeling setup ............................................................................................................................................................ 101
2.5.2.
Structural resonances .................................................................................................................................................. 102
I This research is performed in close cooperation with the Physics of Fluids group of Prof. Dr. Detlef Lohse of the University of Twente.
E-mail address: [email protected]
0370-1573/$ – see front matter © 2010 Elsevier B.V. All rights reserved.
doi:10.1016/j.physrep.2010.03.003
78
3.
4.
5.
6.
H. Wijshoff / Physics Reports 491 (2010) 77–177
2.6.
Concluding remarks.................................................................................................................................................................... 104
Channel acoustics.................................................................................................................................................................................... 104
3.1.
Narrow channel theory .............................................................................................................................................................. 104
3.1.1.
Governing equations ................................................................................................................................................... 104
3.1.2.
Frequency characteristics............................................................................................................................................ 105
3.1.3.
Traveling wave principle ............................................................................................................................................. 108
3.2.
Nozzle boundary ......................................................................................................................................................................... 109
3.2.1.
Nozzle pressure............................................................................................................................................................ 109
3.2.2.
Drop size and speed..................................................................................................................................................... 111
3.3.
Cross-talk..................................................................................................................................................................................... 112
3.3.1.
Local cross-talk ............................................................................................................................................................ 112
3.3.2.
Printhead resonances .................................................................................................................................................. 113
3.3.3.
Acoustic cross-talk....................................................................................................................................................... 114
3.4.
Residual vibrations ..................................................................................................................................................................... 118
3.4.1.
Refill.............................................................................................................................................................................. 118
3.4.2.
Drop-on-demand frequency oscillations ................................................................................................................... 119
3.5.
Concluding remarks.................................................................................................................................................................... 121
Drop dynamics ........................................................................................................................................................................................ 121
4.1.
Drop formation ........................................................................................................................................................................... 121
4.1.1.
Drop shape and properties.......................................................................................................................................... 121
4.1.2.
Impact on acoustics ..................................................................................................................................................... 124
4.2.
Break-off mechanism ................................................................................................................................................................. 126
4.2.1.
Formation of a tail........................................................................................................................................................ 126
4.2.2.
Pinching off .................................................................................................................................................................. 127
4.2.3.
Tail-end speed.............................................................................................................................................................. 129
4.3.
Satellite drop formation ............................................................................................................................................................. 130
4.3.1.
Mist of droplets............................................................................................................................................................ 130
4.3.2.
Rayleigh breakup ......................................................................................................................................................... 132
4.3.3.
Fast satellites................................................................................................................................................................ 134
4.3.4.
Slow satellites .............................................................................................................................................................. 136
4.4.
Drop size modulation ................................................................................................................................................................. 136
4.4.1.
Pulse width and fill-before-fire-level ......................................................................................................................... 136
4.4.2.
Acoustic resonances .................................................................................................................................................... 137
4.4.3.
Break pulses ................................................................................................................................................................. 139
4.4.4.
Meniscus and drop formation oscillations................................................................................................................. 140
4.5.
Concluding remarks.................................................................................................................................................................... 140
Wetting dynamics ................................................................................................................................................................................... 141
5.1.
Wetting of the nozzle plate ........................................................................................................................................................ 141
5.1.1.
Origin of wetting.......................................................................................................................................................... 141
5.1.2.
Wetting regimes and visualization............................................................................................................................. 143
5.2.
A wetted nozzle .......................................................................................................................................................................... 143
5.2.1.
Jetting nozzle ............................................................................................................................................................... 143
5.2.2.
Non-jetting nozzle ....................................................................................................................................................... 146
5.3.
A wetted nozzle plate ................................................................................................................................................................. 147
5.3.1.
Long-range phenomena .............................................................................................................................................. 147
5.3.2.
Driving mechanisms .................................................................................................................................................... 148
5.3.3.
Complete wetting ........................................................................................................................................................ 151
5.3.4.
Marangoni flow............................................................................................................................................................ 152
5.4.
Impact on drop formation .......................................................................................................................................................... 152
5.4.1.
Drop properties............................................................................................................................................................ 152
5.4.2.
Channel acoustics ........................................................................................................................................................ 153
5.5.
Concluding remarks.................................................................................................................................................................... 153
Bubble dynamics ..................................................................................................................................................................................... 154
6.1.
Stability........................................................................................................................................................................................ 154
6.1.1.
Dirt and air entrapment .............................................................................................................................................. 154
6.1.2.
Acoustic detection of air bubbles................................................................................................................................ 155
6.2.
Air entrainment .......................................................................................................................................................................... 155
6.2.1.
Wetting layer ............................................................................................................................................................... 155
6.2.2.
Small dirt particles....................................................................................................................................................... 156
6.3.
The oscillating bubble................................................................................................................................................................. 159
6.3.1.
Size dynamics............................................................................................................................................................... 159
6.3.2.
Impact on channel acoustics ....................................................................................................................................... 160
6.3.3.
Impact on drop formation ........................................................................................................................................... 161
6.4.
The moving bubble ..................................................................................................................................................................... 162
6.4.1.
Balance of forces .......................................................................................................................................................... 162
6.4.2.
Net displacements ....................................................................................................................................................... 163
H. Wijshoff / Physics Reports 491 (2010) 77–177
79
6.5.
7.
The growing bubble .................................................................................................................................................................... 164
6.5.1.
Rectified diffusion and dissolution ............................................................................................................................. 164
6.5.2.
Influence of the actuation ........................................................................................................................................... 165
6.5.3.
Impact on the actuation .............................................................................................................................................. 166
6.6.
Concluding remarks.................................................................................................................................................................... 167
Conclusions and outlook......................................................................................................................................................................... 168
Acknowledgements................................................................................................................................................................................. 169
References................................................................................................................................................................................................ 169
1. Introduction
In this section an overview is presented of the main inkjet developments to clarify the importance of inkjet technology
as a key technology for today’s industry. Besides printing onto paper, many other applications have emerged in the last few
years. The requirements to meet the needs in many areas justify an intensive research program. The basic printing principles
are outlined and the main printhead operation issues are presented. Finally a guide through the next sections is given.
1.1. Historical overview on inkjet technology
Inkjet printing is an important technology in color document production [1]. The rapid development of inkjet technology
started off around the late 1950s. Since then, many inkjet devices have seen the light of day. In this overview, the
attention is mainly restricted to the development towards the most important inkjet concepts of today, namely continuous,
piezoelectric, and thermal inkjet.
The first inkjet-like recording device, using electrostatic forces, was invented in 1858 by William Thomson, later Lord
Kelvin. This was the Siphon recorder as shown in Fig. 1. The apparatus was used for automatic recordings of telegraph
messages and was patented in 1867 (UK Patent 2147/1867). A siphon produces a continuous stream of ink onto a moving
web of paper and a driving signal moves the siphon horizontally back and forth. The first experiments on manipulating
a stream of droplets even goes back to 1749. That year, Abbé Nollet published his investigations on the effects of static
electricity on a drop stream [2,3].
In 1822 the equations to describe the motion of fluids were formulated by the French engineer and physicist Claude
Navier [4], seventy years after Euler had published his equation for ideal liquids without viscosity [5]. Navier also formulated
the general theory of elasticity in a mathematically usable form in 1821 [6]. Hooke published his law on linear elasticity in
1678. George Stokes introduced his equations for the motion of liquids in 1845 [7]. Hence the name Navier–Stokes equations
for the application of classical mechanics to a continuum under the assumption of a stress that is linear with the strain rate.
The foundation of modern inkjet technology is attributed to the Belgian physicist Joseph Plateau and the English physicist
Lord Rayleigh. Plateau was the very first to publish on this field in 1843 with his experiments on the decay of a liquid column
in a density-matched surrounding, the so-called Plateau tank [8]. He noticed that perturbations become unstable when their
wavelength is long enough [9]. Only after Hagen published his wrong result [10], did he formulate the right criterion, the
Plateau argument [11]. He derived the relationship of jet diameter to drop size in 1865 and described his work in detail in
1873 [12]. Lord Rayleigh published a series of founding papers starting with ‘‘Instability of jets’’ in 1878 [13–15]. He added
the flow dynamics to the linear analysis of the decay of liquid jets, and found the right over-stretching of the most unstable
breakup mode. His results were in good agreement with the measurements of Savart in 1833 [16], which was in fact the
experimental foundation of the work of Plateau and Rayleigh. Savart was the first one who recognized that the break-up of
liquid jets is governed by laws, independent of the circumstance under which the jet is produced. He used acoustic energy
to form uniform drops.
What was missing in the studies on the behavior of jets was the realization that the surface tension is the driving force
behind drop break-up. In the studies on dripping drops, gravity does play a central role in the formation of drops [17]. In
the oldest reported observations of the behavior of liquid jets by Leonardo da Vinci in 1508 [18], the cohesion of a liquid,
which results in the surface tension, was assumed to have only a contribution in holding the liquid together. Gravity was
assumed to be the force leading to breakup of a liquid jet, as more quantitatively derived in 1686 by Mariotte [19], at about
the same time as Newton published his famous ‘‘Philosophi Naturalis Principia Mathematica’’, with Newton’s second law
and the definition of Newtonian viscosity. The groundwork for the description of the role of surface tension forces as the
driving force behind breakup was laid by Young in 1804 [20] and Laplace in 1805 [21]. They finally recognized that the same
force, which holds the liquid together, is also the driving force behind the break-up of a liquid jet.
Other important events for the development of inkjet technology were in 1861 and 1865, when Maxwell published
his equations, describing the electromagnetic forces [22,23]. The next event was the discovery of the piezoelectric effect
(electricity from an applied mechanical stress) by Pierre and Jacques Curie in 1880 [24]. Their experimental demonstration
consisted of a conclusive measurement of surface charges appearing on specially prepared crystals, which were subjected
to mechanical stress. In 1881, Lippmann deduced mathematically the inverse piezoelectric effect (stress in response to an
applied electric field). The Curie brothers immediately confirmed the existence of this property [25]. In the following years,
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H. Wijshoff / Physics Reports 491 (2010) 77–177
a
b
Fig. 1. (a) The Siphon recorder is the first practical continuous inkjet device. It was used for automatic recordings of telegraph messages and invented by
William Thomson in 1858 (UK Patent 2147/1867). (b) An illustration from Abbé Nollet showing the first experiments on the effect of static electricity on
a drop stream, published in 1749 [2].
10
6
25
8
7
5
1
9
4
3
2
Fig. 2. Drawing of the first continuous inkjet recording device, which was patented in 1952 (US Patent 2566,443). This was the forerunner to the modern
inkjet printing mechanisms.
the twenty natural crystal classes in which piezoelectric effects occur and all possible macroscopic piezoelectric coefficients
were defined [26].
During the 19th century increasingly sophisticated experiments revealed the non-linear dynamics of jet breakup [27–29],
from which for example the surface tension could be deduced [30,31]. The similar shapes were rediscovered several
times [32,33]. Weber added viscosity to the analysis of jet decay [34]. So the main ingredients for nowadays inkjet devices
were known. In 1931, the use of a piezoelectric material as actuator was reported [35]. Still, it took many decades before
applications of the physical principles of drop formation were used in commercial working devices. In 1951, Elmqvist of the
Siemens-Elema company patented the first practical inkjet device (US Patent 2566,433), which was based on the Rayleigh
breakup. Instead of a continuous stream of ink, as with the Siphon recorder, now a continuous stream of droplets was jetted
onto the paper, Fig. 2. This resulted in Elema’s Mingograph [36]. Instead of being an inkjet printer, it was merely a medical
voltage recorder (e.g. for ECG and EEG). The deflection of the drops was driven through analog voltages from a sensor,
quite similar to current seismic apparatus. The dot positioning varied continuously, perpendicular to the paper transport
direction. The jet was formed from a glass capillary with a length of 3 cm and a diameter of 100 µm, ending with a 15 µm
nozzle opening. The breakup of the liquid jet was not synchronized. The Mingograph was capable of recording signals with
a frequency up to 1.25 kHz. This was in fact the first inkjet printing principle coming to the market.
A continuous inkjet emerged in the 1960s [37,38]. In the early 1960s, R.G. Sweet of Stanford University demonstrated that
by applying a pressure wave pattern to an orifice, the ink stream could be broken into droplets of uniform size and spacing
up to frequencies of 120 kHz [39]. With a controlled jet break-up mechanism, the drops could be charged selectively and
reliably as they emerged out of the continuous jet. Electrically conducting ink can be charged inductively. The charged drops
were deflected towards several positions, when passing through an electric field, to form an image on the substrate. The
H. Wijshoff / Physics Reports 491 (2010) 77–177
81
Fig. 3. Classification of inkjet printing technologies, adapted from [41].
uncharged drops were captured by the gutter and re-circulated in the system. This printing process is known as Continuous
Inkjet (CIJ) printing (US Patent 3596,275), with the Inkjet Oscillograph as the first device. This device was elaborated for use
by the Stanford Research Institute (SRI) for inkjet bar coding work for Recognition Equipment Incorporated (REI).
The A.B. Dick Company elaborated Sweet’s invention. With the Lewis–Brown patent in 1964 (US 3298,030), character
printing was enabled. This resulted in 1968 in the Videojet 9600, the first commercial CIJ printing product. A single jet
with a drop frequency of 66 kHz was deflected to 11 raster positions. These activities were later continued in the VideoJet
Technology company. Another company, that was involved in the further development of the multiple drop deflection
technology, was Sharp. They released their Jetpoint printer in 1973. The developments were boosted by the huge research
efforts of IBM in the 1970s, which licensed the technology from A.B. Dick and the Mead company. This resulted in 1976 in
the IBM 6640 [40], a hardcopy-output peripheral application, which printed by deflecting a jet with a drop frequency of
117 kHz to 40 raster positions.
Honeywell filed in 1968 the Sweet–Cummings patent for multi-jet printing (US 3373,437). In this approach, the drops
are not deflected to be deposited on different positions on the paper, but the deflected drop are captured. The patterns are
printed with un-deflected drops from multiple orifices, the binary CIJ principle. The drops are now deflected in the direction
of the paper transport, and not perpendicular as with the multiple CIJ, and only one deflection level is used. In Fig. 3, an
overview is given of the main modern inkjet printing technologies.
The binary deflection was further developed, not only for bar code printing, but also for advertising purposes, with the
so-called DIJIT printer, introduced in 1973 by the Mead company, which purchased the Cummings interests. The DIJIT head
used 512 jets at a spacing of 100 npi with a drop repetition rate of 12 kHz. In 1983, the Mead company became Diconix, after
being purchased by the Eastman Kodak company. In 1988, Diconix merged with Kodak to become Kodak Dayton Operations
and this part was sold in 1993 to Scitex.
In the 1970s, Hertz of the Lund University of Technology in Sweden developed several continuous inkjet techniques,
which enabled gray-scale printing by varying the number of drops per pixel [42]. In 1977, Applicon introduced the first
color inkjet printer, based on the principles of Hertz. His methods were also adopted by Iris Graphics and Stork to produce
high-quality color images for the prepress color hardcopy market [43]. The Scitex Iris Graphics proofer imaged up to 32
drops on a single dot at a resolution of 300 dpi.
Instead of continuously firing drops it is also possible to create drops only when an actuation pulse is provided: dropon-demand. Major advantages of drop-on-demand (DOD) printers over CIJ printers include the fact that there is no need for
complicated hardware for break-off synchronization, charging electrodes, deflection electrodes, guttering, and re-circulation
systems, high pressure ink supplies and complex electronic circuitry. The first pioneering work in that direction was
performed in the late 1940s by Hansell of the Radio Corporation of America (RCA), who invented the first drop-on-demand
device as shown in Fig. 4. By means of a piezoelectric disc, coaxially arranged with an ink-filled conical nozzle, pressure waves
could be generated that caused a spray of ink drops. However, this invention, intended for use as a writing mechanism in a
pioneering RCA facsimile concept, was never developed into a commercial product [1].
The first DOD technique, that really emerged, was the electrostatic pull inkjet in the 1960s. The principle was known
for a long time [44]. The first patent was filed in 1962 by Winston (US 3060,429). The basic working principle comprises
the following. Conductive ink is held in a nozzle by negative pressure. By application of a high voltage pulse to an electrode
located outside the nozzle, a droplet of ink is pulled out [45–47]. By application of the appropriate deflection fields, the
droplet can be located on the substrate as with CIJ. However, now only one droplet at a time could be between the plates,
otherwise the next drops would follow identical paths. With CIJ, multiple selectively charged drops can follow different
paths through the constant homogeneous electric field. Companies developing electrostatic pull inkjet devices were the
Casio, the Teletype, and the Paillard companies. The Inktronic Teletype machine in the late 1960s was marketed by the
Teletype company. With the model 500 Typuter, the Casio company released a printer of this type in 1971, which was based
on a Paillard prototype. Paillard had already stopped his development. In the 1970s, the DOD electrostatic pull principle was
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Fig. 4. Drawing of the first drop-on-demand piezo inkjet device, which was patented in 1950 but not further developed into a commercial product
(US Patent 2512,743). A piezoelectric disc (5) generates pressure waves in the solid cone (1), which cause a spray of ink drops from the nozzle (2).
Fig. 5. Classification of piezo inkjet (PIJ) printhead technologies by the deformation mode used to generate the drops.
abandoned due to poor printing quality and reliability, although research activities for other applications continue until
today [48–50], with many other applications than printing onto paper [51].
Generally, the basis of piezoelectric inkjet (PIJ) printers is attributed to three patents in the 1970s [52]. The common
denominator of these three patents is, as in the first pioneering patent of 1950, the use of a piezoelectrical unit to convert
an electrical driving voltage into a mechanical deformation of a ink chamber, which generates the pressure required for the
drop formation from a nozzle. The first one is that of Zoltan of the Clevite company in 1972 (US Patent 3683,212), proposing
a squeeze mode of operation [53]. In this mode, a hollow tube of piezoelectric material is used. When a voltage is applied
on the piezoelectric material, the ink chamber is squeezed and a drop is forced out of a nozzle.
The second patent of Stemme of Chalmers University in 1973 (US Patent 3747,120) utilizes the bend mode of piezoelectric
operation. In this mode, the bending of a wall of the ink chamber is used to eject a drop [54]. Therefore, the wall is made of a
diaphragm with a piezo ceramic bonded onto it. The bend mode is also referred to as bimorph or unimorph mode. The third
patent of Kyser and Sears of the Silonics company in 1976 (US Patent 3946,398) also used the bend mode operation [55].
Both bend-mode patents were filed in the same year. The minor difference between the two is that Stemme used a flat disc
of piezoelectric material to deform a rear wall of an ink chamber and that Kyser and Sears used a rectangular plate to deform
the roof of an ink chamber. Silonics was the second company to introduce a piezoelectric DOD printer, namely the Quietype
in 1978, which used the bend mode of the Kyser patent. The printhead, which required 150 V driving amplitude, fired drops
with a maximum repetition rate of 3 kHz. The electrical field is applied in the polarization direction of the piezo material.
The deformation in the poling direction is not used, but the deformation perpendicular to the poling direction. Since the
piezo ceramic is bonded onto a passive membrane, the actuator will bend. Obviously, the main discriminator between the
PIJ patents is the dominating deformation mode used of the piezoelectric material, together with the geometry of the ink
channels, Fig. 5.
The patent of Stuart Howkins (US Patent 4459,601) of the Exxon company in 1984, describes the push-mode version. With
the push mode, also referred to as bump mode, a piezoelectric element pushes against an ink chamber wall to deform the ink
H. Wijshoff / Physics Reports 491 (2010) 77–177
83
100
122
124
102
120
123
116
104
106
108
105
110
111
VOLTAGE
SOURCE
HEATER
106
132
103
112
107
130
114
Fig. 6. Sudden steam printing, the first drop-on-demand thermal inkjet device, which was patented in 1965, but not further developed into a commercial
product (US Patent 3179,042). An electric current from the electrodes (102 and 104) passes through a portion of the ink (116). The ink is preheated (132)
nearly to its boiling temperature and the extra heat from electric current generates the steam that ejects the drops from the nozzle.
chamber. The electrical field is applied in the poling direction and the deformation is in the same direction or perpendicular
to the poling direction. Finally, the patent of Fischbeck (US Patent 4584,590) proposed the shear-mode. In the shear mode
the strong shear deformation component in piezoelectric materials is used to deform a ink chamber wall. The electric field
is perpendicular to the polarization direction of the piezo ceramic. These modes complete the now commonly adapted
categorization of the piezo inkjet printhead configurations: the squeeze, push, bend, and shear mode, Fig. 5. In this figure,
a special version of the shear-mode inkjet is shown, the shared-wall principle, where the piezo ceramic is also the channel
plate.
With sudden steam printing, Fig. 6, Naiman from the Sperry Rand Company basically invented another DOD technique,
thermal inkjet printing, in the 1960s. By boiling aqueous ink at certain time instances, a drop of ink could be generated.
The strength of this design clearly was not immediately acknowledged, since the company did not elaborate this idea into
a commercial product. The idea was abandoned until the late 1970s when Canon and Hewlett Packard (HP) picked it up.
In 1979, Ichiro Endo and Toshitami Hara of the Canon company re-invented the drop-on-demand printhead, which is
actuated by a water vapor bubble, called bubblejet. They were both working on a piezo-based drop-on-demand printhead.
Accidentally, Endo watched a spray of ink from a needle, after touching the needle with a hot soldering iron. The first
BubbleJet printer was launched in 1981 and was the first side-shooter device. The droplet was ejected in a perpendicular
direction away from the evaporating bubble. In the same period also HP developed their thermal inkjet technology. John
Vaught and Dave Donald, working on a squeeze mode piezo printhead, both got inspired by the working principle of a
coffee percolator. This led to the first successful low-cost inkjet printer in 1984. The jetting direction was in line with
the evaporating bubble, the so-called top-shooter design. The ThinkJet fired 180 pl drops from 12 nozzles at a maximum
repetition rate of 1.3 kHz. In 1985, Canon introduced the BJ-80, which contained a printhead with 24 nozzles.
The invention of the thermal inkjet (TIJ) fundamentally changed inkjet research. By the replacement of the piezoelectric
by a thermal transducer, the main bottleneck concerning miniaturization was resolved. The thermal transducer became a
simple, small, and cheap resistor. Canon and HP joined their forces and protected themselves with a wall of patents. The real
inkjet revolution started in 1988 with HP’s DeskJet, which fired 85 pl drops at a repetition rate of 3.6 kHz. Canon caught up in
1990 with their BJ10, which had 64 nozzles at a resolution of 360 npi, and in 1992 with the BJ200. In 1991, HP launched the
first full color TIJ. Their PainJet, introduced in 1986, was a three-color printer. Lexmark, spun off from IBM in 1991, joined
the top three of thermal inkjet vendors in 1993 with the ExecJet IJ 4076.
TIJ can be manufactured using mass-production based on IC manufacturing technologies. This made the cost per nozzle
much lower than the cost per nozzle of a piezoelectric printhead. Both the fact that inkjet printers now could be miniaturized
and its low cost of manufacturing made TIJ the superior inkjet technology at that time. HP solved the reliability problem of
the thermal drop-on-demand printheads by the concept of disposable heads and increased the performance of their thermal
printheads continuously as shown in Fig. 7. HP claims that TIJ jets everything that nucleates like toluene, silver suspensions,
and even functional proteins. Currently thermal inkjet, with top-shooters of HP, Canon, Lexmark and Olivetti (with a Canonbased technology) and side-shooters of Canon (the first series) and Xerox, dominates the low-end home/office color printer
market. Xerox started within a few months after the demo of Canon in 1981 of their TIJ research and this became three years
later their only inkjet technology. Océ also applies thermal drop-on-demand inkjet in its wide format color printing systems
with printheads of various manufacturers.
After the introduction and immense success of TIJ, PIJ research efforts were largely diminished. However, critical in TIJ
is the spreading and intercolor bleeding of water based inks. This requires special coatings on the media surface. At high
productivity, cockling and drying of the media is another problem. Therefore, solid inks (hot-melt or phase-change ink),
which require piezo actuation, remained important.
Only a few companies continued their research into PIJ. New initiatives with a bump-mode design were taken in 1983
by Howtek and Exxon, which were later acquired by Dataproducts. The 965 printer had printheads with 32 nozzles, but
the product, although being technologically very impressive, was not a success. Epson emerged as the PIJ leader with the
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Printhead
drops per
second*
1 billion
100 million
10 million
1 million
1 thousand
1980
1985
1990
1995
2000
2005
2010
2015
Fig. 7. HP Moore’s law: inkjet printhead performance as printhead drops per second (the number of nozzles times the maximum drop-on-demand
frequency) doubles every 18 months for the past 20 years. It started with 12 nozzles in a single color glass chip in 1984. Via different printer series (the
different markers), this evolved into the 3900 nozzles in a six color single silicon chip in 2006, the Scalable Printhead Technology (SPT) [56].
introduction of the 12 nozzle SQ-2000 in 1984 and the big success of the Stylus 800, introduced in 1993. The printhead had
4 × 12 nozzles at a resolution of 90 npi each. Now the bump mode actuation is still used by Epson in their MLP heads. The
bump mode is also used by Hitachi (which acquired Dataproducts) and applied in Ricoh printers like the IPSiO and the G707,
by Trident in their PixelJet and the 780Jet, which have the unique capability that the heads can be re-assembled again, and
by Brother in their HS5000 printer.
The bend mode is used by Tektronix (acquired by Xerox) in their pagewide heads in the Phaser printer series, by
Brother/Kyocera in their new pagewide array and by Epson in their MLChips heads, introduced in 1997 for the low-end
Stylus color printers. Epson uses water based inks, but Mimaki, Mutoh and Roland use oil based and solvent inks with Epson
heads. Also Brother used the bend mode in their MCF580 printer.
The shear mode design came into the field in 1984 with Spectra, a Xerox licenses’ formed company, in their Galaxy and
Nova heads. Spectra was acquired by Markem, later by Dimatix and finally by FujiFilm. A special version of the shear mode is
the shared wall design of Xaar. This company started in 1990 for commercializing the work of the Cambridge researchers and
licensed its technology to the Swedish MIT, an IBM spin-off. In 1999, Xaar acquired MIT from Pelikan to become XAARJet,
by then, IBM had sold all its printing activities. Other companies, which use the shared-wall concept, are Brother/Kodak in
their Kodak5260 printer, ToshibaTEC, Konica, Sharp, and MicroFab.
The squeeze mode design never evolved further towards a multi-nozzle printhead for the graphical printing markets.
Single nozzle devices or heads with only a few operating nozzles were developed at several laboratories. Also at Philips
Laboratories in Hamburg [57,58]. In 1981, the P2131 printhead was developed for the Philips P2000T microcomputer.
These activities were later continued as a spin-off company Microdrop Technologies [59]. The printhead is used for several
specialized applications, for example the production of lenses [60].
At present, both TIJ and PIJ printing have evolved into the two most important technologies when it comes to printing. The
initial advantages of TIJ over PIJ have been leveled over the years by further development of the PIJ technology. Thousands
of nozzles can fire drops at several tens kHz with drop volumes down to 1 pl. Epson, which has put in more effort to increase
the number of nozzles in their PIJ printhead, focused on print quality. HP focuses more on productivity. Canon is moving
also towards high quality printing with very small drops. Lexmark, the number four, is somewhat behind, mainly because
of the less R&D the company has spend on inkjet development.
A fundamental strength of the PIJ technology is its ability to deposit a wide variety of materials on various substrates in
well-defined patterns. Recently many other applications than printing onto paper emerged [61–68]. In the display market,
inkjet technology is used to manufacture Flat Panel Displays (FDP), Liquid Crystal Displays (LCD), color filters [69] (a part of
LCDs), Polymer Light Emitting Diodes (PLED) [70], and flexible displays. The accompanying performance criteria are among
the major driving forces behind much research and development efforts [71,72]. Within the chemical market, the inkjet
technology is mainly used as a tool for research purposes. The unique capacity of the technology for dispensing small doses
of liquids makes it very useful for this market. Applications include material and substrate development as well as coating
purposes [73–75].
In the electronic market, inkjet printheads are used to create functional electrical traces using conductive fluids on both
rigid and flexible substrates. One of the first applications of inkjet technology within this field was that for the production of
Printed Circuit Boards (PCB) [76,77]. Other applications include the fabrication of electric components and circuits such as
Radio Frequency Identification (RFID) tags, wearable electronics, solar cells [78], fuel cells, and batteries. Challenges for the
inkjet technology within this field include the spreading of the ink and the required guarantees of continuity of the jetted
lines and interconnections [79–85].
Three-dimensional mechanical printing claims the inkjet technology as a tool for rapid prototyping, small volume
production, and the production of small sensors [86–88]. Jetting of UV-curable optical polymers is a key technology for
the cost-effective production of micro-lenses. These tiny lenses are used in devices from fiber optic collimators to medical
systems. The ability of inkjet technology to precisely jet spheres in variable, but consistent, drop sizes provides opportunities
for the cost reduction of existing optical components and innovative new designs [89–91].
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The life science market is rapidly expanding with new requirements for precise dispensing of DNA and protein
substances [92]. The high costs of these fluids make inkjet technology with its precision placement and tight flow control an
excellent dispensing tool. Applications include the use for DNA research, various medical purposes such as dosing of drugs,
and food science. A quite futuristic application is the use of inkjet printing for the fabrication of living tissue [93–97].
1.2. Printing principles
The graphic printing applications require certain performance criteria to be met. For an inkjet printhead, an important
set of requirements is related to the resulting drop properties, namely:
• Drop-speed: the resulting droplets are required to have a certain speed, typically several m/s. A high drop speed results
•
•
•
•
in a short time of flight. The sensitivity for disturbing influences like variations in the printhead–substrate distance will
be less, thus the dot position errors will be smaller.
Drop-volume: depending on the application under consideration, the performance requirement concerning volume
typically varies now from 2 to 32 pl. For some applications, it is required that the drop-size can be varied during operation.
For example, for large areas that need to be covered large drops are desired, whereas for high resolution printing small
drops are desirable. This is referred to as drop-size modulation.
Drop speed and volume consistency: the variations in drop volume and speed must stay within a certain percentage
band, typically around 2%. This is to avoid irregularities in the printed object.
Drop shape: the shape of the dots on a substrate is negatively influenced by the formation of tails or satellite drops. These
are highly undesirable for the quality of the print.
Jet straightness: the droplets have to be deposed in a straight line towards the substrate, typically within 10 mrad
accuracy.
Productivity and stability are important requirements, which are closely related to the jetting process. The productivity
of an inkjet printhead is mainly determined by the jetting frequency, defined as the number of drops that a channel jets
within a certain time fdod , and by the number of nozzles Nnoz , defined by the integration density or nozzles per inch and the
printhead width. A high integration density has a lot of consequences for the functionality and the producibility. For the
productivity of an inkjet printer, the firing power Pjet of the printheads is the key number. The productivity in m2 /s is given
by:
Pjet =
Nnoz fdod
dpi2
× (24.4 · 10−3 )2
(1)
with dpi the number of dots per inch [98]. This results in a print time tprint for an area Aprint of a printer of:
tprint =
Aprint
(2)
ηPjet
with η the efficiency of the printer, which plays a very important role in a scanning system. Most inkjet printers use a carriage
with several printheads which moves over the full width of the paper. Between the strokes of the carriage, or print swaths,
the paper is transported over a certain distance to cover the full area. At higher frequencies, the carriage turn-time and the
paper step give more and more limitations. These limitations result in an optimum DOD frequency for the productivity of
scanning printer systems [99]. An expression for the optimum DOD frequency fm , below which print duration dominates and
above turning duration, is derived with a model which includes many parameters such as carriage driving force Fc , carriage
mass mc and width xc , print width xp and resolution rx , and multi passing px in the swath direction, i.e. the number of strokes
to print one line:
fm =
rx
px
s
Fc (xp + xc )
2mc
.
(3)
For productivity the maximum applicable DOD frequency for 600 dpi prints in most scanning printer concepts is 30–40 kHz.
High productivity single pass printing requires a page-wide array with thousands of nozzles as explored by Spectra [100,
101] and Brother-Kyocera [102], and used by Xerox/Tektronix in their Phaser printer series. Another way to increase the
number of nozzles is to use multiple nozzles per channel as explored by Trident [103].
The number of dots per inch is important for print cost and print quality. Print cost is directly related to the thickness of
the ink layer or amount of ink/m2 . As a first order approximation, the required drop volume for a 600 dpi dot is about 32 pl
and for a 1200 dpi dot about 4 pl, since volume scales as the third power of the spatial resolution. The total volume for a
600 dpi dot with four 1200 dpi drops becomes 16 pl instead of 32 pl with a single 600 dpi drop. Print quality is better with
smaller dots at a high resolution. Finer details can be represented and also the graininess of the print is much less with small
dots. For water based inks the lower amount of ink with smaller drops results in shorter drying time, another drive to move
towards smaller drops. So there is a trade-off between productivity and print quality/cost. Drop size modulation is a way to
meet both requirements. The small drops are used for achieving a high print quality and the large drops for productivity.
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Fig. 8. 3D CAD drawing of a printheadprototype, developed at Océ, showing (a) the melting unit, (b) the filter units, (c) the reservoir, (d) the static pressure
hose, (e) the central part, and (f) the electronic driving supply.
The stability of the jetting process is one of the most important performance requirements for inkjet printheads. Typically,
it is required that at most one failure occurs per certain number of jetted drops. For printing onto paper this is typical one
billion, but in some industrial applications no failure at all is allowed. Additional, more general, requirements include the
lifespan of the printhead (typically more than ten billion actuations per channel, a fundamental strength of PIJ over TIJ), the
materials compatibility (a wide variety of inks must be depositable, again a strength of PIJ over TIJ), the maintainability, and
the cost of production and manufacturability of the printhead (a weakness of PIJ).
1.3. Printhead operation
1.3.1. Working principle
The starting point for the review is a printhead, developed at the R&D department of Océ, This printhead is referred to
as the reference printhead. The final goal of the printhead operation is firing billions of drops. Before a drop is fired, a lot of
processes must take place. In the case of hotmelt inks, it starts with the melting of the ink in the melting unit (a), Fig. 8. This
unit must have enough capacity. A good heat transfer and draining of the melted ink at a given maximum temperature like
135 °C is necessary, taking into account some overshoot and time delay. Another task is filtering (b) the ink. The next unit
is the reservoir (c), which must have enough volume, while keeping the total printhead dimensions within certain limits.
Important aspects concerning the reservoir are:
•
•
•
•
•
filtering with non-woven filters with high dirt holding capacity
removal of air bubbles
temperature control
closing static pressure, for example supplied with hose (d)
ink-level sensing.
The lower part of the printhead is the central part (e) where drop formation takes place. This part is the subject, which
forms the basis of the discussion in this review and is shown in more detail in Fig. 9. The main components in the central
part are:
•
•
•
•
•
a last filter (k), to remove the dirt particles from the ink
the channel block (h), in which the ink channels (l) are made
the nozzle plate (g), where the drops are formed
the actuator foil (i), which covers the ink channels in the channel block. The foil is also connected to:
the actuator plate with piezo elements and substrate (j). Here the driving force for the drop formation process is
generated. The required electric voltage is supplied by the electronic flex (f), Fig. 8.
The ink path from the reservoir and last filter to the nozzle can be divided into a supply channel, a pressure channel
and a connection channel. The actuation takes place in the pressure channel, which is shown in Fig. 9 in a front view of a
channel. When a voltage is applied on a piezo element, this element will change its shape. This deforms the ink channel,
which generates the pressure waves to fire a drop. The reaction force is supplied by the substrate, which is connected to the
channel block via passive piezo elements.
H. Wijshoff / Physics Reports 491 (2010) 77–177
87
Fig. 9. Side view of channel structure of the reference printhead and front view with actuation principle as explained in Section 2.2.1, with (c) the reservoir,
(g) the nozzle plate, (h) the channel block, (i) the actuator foil, (j) the actuator with piezo elements and the substrate, (k) a last filter, and (l) the ink channel.
Fig. 10. Schematic drawing of the actuation principle. An electric voltage on a piezo element enlarges the channel and a negative pressure is generated.
After reflection at the reservoir this becomes a positive pressure wave (only the part of the wave, which drives out a droplet, is shown). The positive
pressure wave is amplified by the second slope of the driving waveform to get a large positive pressure peak at the nozzle, which fires a drop.
Typical dimensions are length 5–20 mm for the total ink channel with a cross-section of 0.01–0.05 mm2 . The channel
block material is graphite with shaped channels, or a metal like brass or silicon with etched channels. The actuator foil has
a thickness of 5–50 µm and we use poly-imide, metal, glass, or silicon as the material. The actuator plate is a piezo-ceramic
material with a height of typical 500 µm with diced elements. In most cases we use a substrate to support the element
structure. The substrate is several millimeters longer than the piezo elements, to enable the connection of the electronic
flex. The nozzle plate thickness ranges from 30–125 µm and nozzle diameters from 18–50 µm. Different nozzle shapes are
made in nozzle plates of nickel, tantalum, poly-imide or silicon.
A long ink channel with a nozzle at the right side and a large reservoir at the left side is the simplified geometry of the
inkjet device like in [104], Fig. 10. 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 piezoelectric effect. This results in pressure waves
inside the channel. The pressure waves propagate in the channel direction and will be reflected when the characteristic
acoustic impedance Z of the channel changes. The acoustic impedance of the channel depends on the size of the channel
cross-section A and the speed of sound c as:
Z =
ρc
(4)
A
with ρ the density of the ink. The speed of sound is influenced by the compliance of the channel cross-section, see
Section 3.1.3. The reflection and transmission coefficients at the interface between domain 1 and domain 2 are:
R=
Z2 − Z1
Z1 + Z2
T =
2Z2
Z1 + Z2
.
(5)
When the compliance does not change, the following relationship holds:
R=
A1 − A2
A1 + A2
T =
2A1
A1 + A2
(6)
At the large reservoir (A2 A1 ) the transmission coefficient is zero and the reflection coefficient equals −1. This means
that the pressure wave will be completely reflected and the amplitude of the wave will change.
The charging of the piezo element (a) enlarges the channel cross-section and the resulting negative pressure wave will be
reflected at the reservoir at the left (b). The large reservoir acts as an open-end and the acoustic wave returns as a positive
pressure wave (c). The de-charging of the piezo element reduces the channel cross-section to its original size. This will
amplify the positive pressure wave when tuned to the travel time of this acoustic wave (d). The channel structure and driving
pulse are designed to get a large incoming positive pressure peak at the nozzle (e), which drives the ink through the nozzle.
Acceleration of the ink movement in the small cross-section of the nozzle (conservation of mass and incompressibility)
results in drop formation.
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Fig. 11. Outline of experimental setup with printhead control, driving electronics and optical recording equipment.
1.3.2. Printhead testing
Droplets are measured by means of optical methods [105,106]. Especially the instabilities of a liquid jet are difficult to
measure [107–109] and a number of optical techniques are used for quantitative measurements [110–112]. In our research
we used stroboscopic illumination at drop formation rate and high-speed camera recordings up to 160,000 images/s with
a Phantom V7 camera from Vision Research or up to one million images/s with a Shimadzu HPV-1 camera. The basic setup
for the optical measurements is outlined in Fig. 11. The setup can be divided into a part which controls the printhead and a
part to visualize the droplets.
The required reference temperature is reached by a PID controller (Eurotherm 2408), which measures the temperature
of the printhead with thermocouples and controls the input voltages for the heating elements. The printhead is mounted
in a vertical direction with the nozzles face down, similar to its position in an inkjet printer. To avoid that the ink simply
flows out of the nozzles under the influence of gravity, an air pressure unit (TS 9150G) makes sure that the pressure in the
ink reservoir remains 8 mbar below the ambient pressure. The setup is connected to a personal computer that is equipped
with National Instruments IMAQ PCI 1409 and PCI GPIB cards for image capturing and processing and for communication,
respectively. A Newport MM3000 motion controller is used for automatic positioning of the printheads in the measuring
positions with Newport xy-tables.
We use Labview software to control the measurements and the measuring results can be directly exported to for example
Excel for further analysis. After defining the actuation signal, it is sent to an arbitrary waveform generator (Philips PM
5150/Wavetek 75A). The waveform generator sends the signal to an amplifier (Krohn-Hite 7602). From the amplifier, the
signal is fed to a so-called switch-board. The switch-board is controlled by the personal computer and determines which
channels are provided with the appropriate actuation signals.
We use a standard CCD camera (for example Sony SSC-M370CE or Watek LCL902K) to capture the images from an
Olympus SZH-10 microscope of the drop formation with a frame rate of 25 images/s. A LED produces flashes of 100 ns
and the strobe frequency is the same as the drop repetition rate or DOD frequency. We will only see the reproducible part of
the drop formation, because the images are integrated over many (depending on the DOD frequency up to several hundred)
droplets, recorded at a certain time interval after the start of the actuation [113,114]. Drop properties can be derived very
easily from these images. The drop speed is calculated from the distance of adjacent drops and the known repetition time.
The drop size is calculated from the number of pixels, or by weighting a certain number of captured drops. Drop direction
is calculated from the centers of mass. To capture non-reproducible phenomena we use the stroboscopic illumination at
camera rate or a high-speed camera recording [115]. With the first option every 40 ms one drop is visualized at a certain
time after the start of the actuation with a progressive scan camera. With a high-speed camera all drops are recorded with
1–10 µs time intervals, in a certain time frame (up to typical 1 s), using a trigger signal to read back the relevant time frame.
Another option is the use of laser-Doppler interferometry (Ono Sokki laser vibrometer LV1500) to record the meniscus
movements without drop formation. The measuring principle is based on mixing an undisturbed and a Doppler-shifted laser
beam. The Doppler effect, the frequency shift ∆f , of the reflected beam is given by:
1f =
2v
c
f0 =
2v
λ
(7)
H. Wijshoff / Physics Reports 491 (2010) 77–177
89
Fig. 12. Outline of Paint measurement. Switching the piezo elements between an electronic driving circuit and a measuring circuit enables both the
actuation of the channels and the measurement of the pressure variation inside the channels.
with f0 the frequency of the undisturbed laser beam, λ the wavelength of the laser beam, c the speed of light, and v the
normal speed of the surface which reflects the laser beam. The meniscus speed is derived from the interference pattern of the
reflected beam with the reference beam and the frequency characteristics are recorded with an HP 3585A spectrum analyzer.
With laser-Doppler it is also possible to measure actuator displacements with an accuracy of 20 nm [116]. Another setup for
the latter is a speckle interferometer. The speed of a non-jetting meniscus can also be derived from the stroboscopic CCDcamera recordings, when zooming on the nozzle exit area [117]. Another alternative is the built-in measurement method
of the meniscus position, which also allows measurements in the jetting situation. This is offered by a sensing nozzleplate
with a capacitive layer inside the nozzle [118,119].
All these measurements give details on the ink flow outside the printhead and the deformations of the exterior of the
printhead. The phenomena inside the channels are difficult to measure. Only when using special transparent printheads,
e.g. channels in or covered by a glass plate, and flow tracing particles can the flow inside be measured [120]. The only
suitable method for the opaque heads uses the actuator also as a sensor. As is generally known, a piezo can be used as an
actuator or as sensor, see e.g. [121]. For that, one uses the piezo’s inverse (actuator) and direct (sensor) piezoelectric effect.
The former comprises the following. If an electrical voltage V is applied to the piezo unit, a displacement y of the piezo unit
results. The latter refers to the following phenomenon. If a force F is applied to a piezo’s surface, an electric charge Q results.
Together, this behavior can be described as:
y
Q
d
=
C
1/k
d
V
F
(8)
with C the capacity, d the piezoelectric charge constant, and k the stiffness of the piezo, see Section 2 for more details.
Switching the piezo elements from the electronic driving circuit to a measuring circuit gives an accurate recording of
the average pressure inside the ink channel, which we from now on call the ‘‘Paint’’ signal (Piezo-Acoustic sensing of INk
channels in the Time domain), Fig. 12. First the driving waveform is applied, which takes 5–20 µs. After that, the current from
the piezo element can be measured until the next actuation cycle starts. The main problem for this setup is that the amplitude
of the Paint signal is only 50–100 µA while the current for charging the piezo is about 10 mA. The big discharge current from
the piezo can disturb the measurement, so the piezo must be completely de-charged before the acoustic measurement starts.
However, with a reference a Paint measurement during actuation is even possible. One option is to calculate the
contribution of the direct path to the Paint signal and subtract this from the measurement [122]. The main drawback of
the computational compensation is related to the required accuracy of the piezo model. The non-linear behavior of the
piezo elements is very difficult to model accurately.
The acoustic measurement also enables monitoring of jetting stability [123], see Section 6 and feed-forward control of
the driving waveform. For most designs, the input waveform is manually shaped, based on physical insight into the working
of a printhead. Normally, the actuation pulse is tuned to the first eigenfrequency of the ink channel. Additionally, somewhat
more complex waveforms are designed for purposes like smaller droplets and damping of the residual vibrations. A control
framework enables the systematic exploration of better driving waveforms to enhance the performance of the printhead,
without having to perform a redesign of the printheads. Iterative Learning Control (ILC) as feed-forward control is very
effective in improving the performance of a process that performs repetitive tasks [124–126]. So more specifically, given
the highly repetitive character of the jetting process, ILC was a logical choice as control strategy for our printheads [122].
Calculations of feed-forward signals require a good understanding of the complete system dynamics.
With measurements we have only limited access to the interior of the printhead. We need not only for control purposes
more information on the phenomena preceding the drop formation for a better understanding of the operating principles of
the piezo printhead. Details on ink flow and acoustic pressure waves are only available through modeling [127]. Therefore,
the modeling of the physical phenomena with available commercial codes and the development of dedicated special models
are an essential part in the development of a new inkjet technology. Added to the measurements, the modeling reveals the
chain of processes, which lead to drop formation. This enables a faster and better development of new printheads [128].
When modeling inkjet printheads, we have to face many challenges, like with the modeling of all Micro ElectroMechanical Systems (MEMS) [129]. The modeling comprises a multi-scale simulation from nanometers to meters of multiphysics: solid mechanics, fluid dynamics, electromagnetism, materials science, electronic circuit design, mechatronics etc.
Most of the models used in our research are derived from continuum mechanics, although in some cases the continuum
mechanics are pushed to its limits. The continuum theory requires that the variables like density, pressure and velocity
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are defined by some averaging process and be determined as the solution to some system of equations. We must keep
this in mind when going to the smallest length scales. The domains of interest in our research are the structural dynamics,
including piezoelectricity and acousto-elastic interaction, and the fluid dynamics [130]. The management of small-scale
flows is a common denominator in micro-electromechanical systems (MEMS) [131].
1.4. Guide through the sections
The aim of the research is to explore the processes which lead to the final goal, the stable and reproducible formation of
billions of drops. The first step is the actuation. The transformation of an electric voltage to a deformation of the ink channels
is described in Section 2. The deformations inside the printhead cannot be measured and modeling with the commercial
finite element code Ansys plays an important role. The deformation of the channel cross-section results in pressure waves.
The acoustic properties of the ink channels are the subject of Section 3. The pressure waves play a central role in the printhead
operation. They connect the electrical/mechanical domain of the actuator to the fluid dynamic domain inside and outside
the nozzles. Measurements are done with the laser-Doppler setup and the acoustic Paint measurement. Details on ink flow
and acoustic pressure waves inside the ink channel are available through modeling. The acousto-elastic interaction plays an
important role in the ‘‘narrow-channel’’ model. Acoustic modeling with Ansys gives the details of the interactions with the
structure dynamics of the printhead. A finite volume method, based on traveling plane waves, provides an efficient model
for the refill of the nozzle and the influence of residual vibrations. A lumped parameter model provides an efficient method
for modeling the acoustic properties of small MEMS-based inkjet printheads.
Another approach, modeling the complete printhead operation with one model in a multi-physics approach [132], leads
to a very long calculation time, even when for example the structural part is modeled with lumped elements [133], and is
not further considered in this review.
The drop formation is the final goal of the printhead operation. The pressure waves in the ink channels are the driving
force behind the drop formation process. In Section 4, the details of the drop formation process are described. Optical
measurements and modeling provide the information we need. For the modeling of the free surface flow with surface
tension, and its impact on channel acoustics, we use the commercial volume of fluid code Flow3D [134]. The main drop
properties can also be derived with a simple model based on a balance of energies and empirical relationships. The pinchoff process itself can be described with scaling laws.
Also the flow phenomena on the nozzle plate are important. Wetting of the nozzle plate can influence the drop
formation process. The material interactions, which determine the wetting properties, are not known in detail. Therefore,
an experimental study of the wetting phenomena is performed. This is described in Section 5. Simple analytical formulas
can describe some aspects, but real material interactions can only be resolved through molecular modeling, which still
has to be done yet. Wetting can also result in air entrainment. Air bubbles play an important role in the jetting stability.
The theoretical and experimental research on the generation and the behavior of air bubbles and their impact on the
printhead performance is the subject of Section 6. The acoustic measurement is an indirect measuring tool for the existence
of air bubbles. Transparent heads are developed for direct visual measurements. Modeling is done with different numerical
models.
In the last section the results are summarized, conclusions are drawn and an outlook on further developments is
presented. The experiments are done with printheads developed at Océ Technologies B.V. For the research, a transparent
test ink is used as reference fluid, which has a viscosity of about 10 mPa.s at the jetting temperature of 130 °C, a surface
tension of 30 mN/m, a density of 1000 kg/m2 and a speed of sound of 1250 m/s.
2. Structure dynamics
In this section the driving force in piezoelectric printheads is discussed. The actuator design for the reference printhead
will be presented, which is based on the bump mode actuation, but also the alternatives, the bend mode and the shear mode
actuation, will be discussed. The deformation of the channels to generate the pressure waves for firing droplets from the
nozzles can also result in local cross-talk effects, like the direct and the pressure induced cross-talk effect, as will be shown
with numerical simulations. Exciting many channels at the same time excites also resonances in the printhead structure.
2.1. Piezoelectricity
The driving force to fire a droplet with a piezo inkjet printhead is generated by the actuator, which deforms the structure
through the inverse piezoelectric effect. Barium titanate (BaTiO3 ), the first piezoelectric ceramic with a perovskite structure
(a tetragonal/rhombohedral structure very close to cubic), was found around 1943. Roberts detected the piezoelectric effect
in BaTiO3 in 1947. In 1954, the discovery of the piezoelectric ceramic Pb(Zrx Ti1−x )O3 , lead zirconate titanate (PZT), was
reported by Jaffe (US Patent 2708,244). In the following years, PZT became the main industrial product in piezoelectric
ceramic materials [135]. Many actuator applications have emerged for the use of PZT [136,137].
Ceramic perovskites have a cubic structure that is stable at temperatures above their Curie temperature, as seen in Fig. 13.
When the temperature decreases and falls below the Curie temperature, the structure changes. The cubic structure becomes
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H. Wijshoff / Physics Reports 491 (2010) 77–177
a
b1
b2
b3
c1
c2
c3
Fig. 84. High speed camera recordings with (a) the undisturbed drop formation in the left picture. The drop formation with partial blocked nozzles are
shown in the other pictures, (b) with drops still jetted away and (c) without jetted drop. The recordings are made with a Phantom V7 high speed camera
at 60 µs after the start of the actuation.
The drop formation is even blocked when the ink layer becomes more than 20 µm. A jetting nozzle induces several flow
patterns in the ink layer. The continuous jetting of drops induces an air flow towards the nozzle. This air flow drags the
outer part of the ink layer always towards the nozzle. As a consequence a strong sink flow can be visible or a Couette type of
flow, when the overfilling of the nozzle adds ink to the inner part of the ink layer. Also an actuated, but non-jetting, nozzle
induces all kind of flow phenomena. Dipole, source and sink flow patterns complete the observed phase diagram of possible
flow patterns in the ink layer around an actuated nozzle.
The nickel nozzle plate and the ink are in a complete wetting regime. Older ink on the nozzle plate has other properties
which results in a partial wetting regime. This older ink accumulates in the middle of the nozzle plate. The dynamic
movement of the ink in an actuated nozzle results also in an increase of the surface tension. The precursor film of fresh
ink, which was measured to be 13 nm in thickness, provides a way of communication between the nozzles and the central
band of ink. A trunk flow drives the ink from the middle of the nozzle plate towards actuated nozzles. Marangoni flow is the
driving mechanism behind this flow pattern.
An ink layer can also have a negative impact on the jetting stability. This will be discussed in the next section.
6. Bubble dynamics
In this section, the main factors which determine the jetting stability are discussed. Even after removing large dirt
particles and air bubbles from the ink, there are sometimes air bubbles generated during the drop formation process. We
will see that because of the relative low acoustic pressures in the printheads, air entrainment is the only mechanism for
the generation of air bubbles. Wetting and small dirt particles play a crucial role in this process. There is a critical ink layer
thickness, which results directly in air entrainment. Dirt particles with a size of 20 µm disturb the drop formation and can
also induce the generation of air bubbles. Air bubbles can be measured directly in special transparent heads. The effect of
air bubbles on the channel acoustics can be measured by using the piezo actuator also as a sensor. Then it will be discussed
that a small air bubble oscillates at its natural frequency after the collapse. The influence of the confined space, i.e. the ink
channel with nozzle, is that the collapse and the after-bounce are suppressed. If an air bubble can dissolve or be jetted out,
no problem is generated. However, at the end of this section, we will see that an air bubble can also grow in an acoustic field
by rectified diffusion. The air bubbles grow to a size comparable to the displacement by the acoustic pressure waves. A large
air bubble will then counteract the pressure build-up and the drop formation process ceases completely.
6.1. Stability
6.1.1. Dirt and air entrapment
A very important requirement for todays productive drop-on-demand inkjet printers is the stability of the jetting process.
Large dirt particles with a radius more than 15 µm can block a nozzle opening completely and there will be no drop formation
at all. A nozzle can also be partially blocked by a large dirt particle. This results in severe deviations of the drop speed and
size, the jetting angles and the drop shape, see Fig. 84. Big dirt particles can originate from the head itself (e.g. graphite
particles), dragged along with the ink (e.g. not good enough filtering or non-compatible ink components), or can come from
outside (e.g. paper fibres and common dust). The large dirt particles have to be removed during normal operating conditions.
The most common cause of instabilities which remains are air bubbles.
Air bubbles will influence the acoustics and the drop formation process and can result in nozzle failure [478,479]. The ink
can contain a lot of air bubbles before the drop formation starts. This air can originate from the ink manufacturing process
H. Wijshoff / Physics Reports 491 (2010) 77–177
155
or from failures in the ink supply system. For a normal printhead operation, these sources have to be eliminated and further
research is therefore concentrated on the generation of air bubbles during normal printhead operation. The removal of air
is also very important in experiments with dripping drops [262,480].
Two possible sources for the generation of air bubbles during normal printhead operation are cavitation [481] and
air entrainment at the nozzle. Cavitation takes place when the negative acoustic pressure exceeds the Blake threshold at
the cavitation nuclei. Cavitation will take place somewhere inside the ink channel where the negative acoustic pressure
amplitude has its largest value. Cavitation can take place at small air bubbles, at non-wetted channel/nozzle walls, at nonwetted dirt particles, or at inclusions by surfactant [482]. So, for a stable drop formation, the ink must be free of air and
dirt, the channel walls must have good wetting properties, and when using surfactant, the concentration must stay below
the critical micelle concentration. Then, the acoustic design of our printheads can be made such that the acoustic negative
pressure remains always above the threshold for cavitation, see also Section 3. Then, air entrainment at the nozzle during
printhead operation remains as the only possible source for the generation of air bubbles.
6.1.2. Acoustic detection of air bubbles
The disturbance or the failure of the drop formation process itself is not suitable for a quantitative measurement of the
behavior of the bubble itself. The printheads developed for application in a printer are opaque. A direct observation of air
bubbles in these printheads is not possible. Only in a special transparent head is a direct observation of air bubbles possible
[483–485]. A way to measure the effect of air bubbles in the opaque heads is to use the piezo actuator also as a sensor. This
is the Paint measurement, see Section 1.3.2. With the Paint measurement, the electric current from the piezo elements is
measured. The Paint current Ip from a piezo element with area Ap is given by the equation:
Ip =
dQ
dt
= Ap
dD3
(130)
dt
with D3 the electric displacement or charge density in the polarization and actuation direction of the piezo element, see
Section 2.2. The electric field is in the actuation direction E3 = V /hp , with hp the height of the piezo element. With Eq. (13)
we get:
Ip = d33 Ap
dT3
dt
+ 3 A p
dE3
dt
.
(131)
The normal stress component T3 is generated by the pressure P in the ink channel. With V the voltage on the electrodes,
this equation becomes:
Ip = d33 Ap
dP
dt
+
3 Ap dV
hp
dt
.
(132)
The capacitance of the piezo element is given by Cp = 3 Ap /hp . After integrating the pressure, which can show local
variations, over the length lp of the piezo element we get:
Ip = Cp
dV
dt
lp
Z
+ d33 bp
0
dP
dt
dz
(133)
with bp the width of the piezo element. The Paint current is now known in terms of the actuator voltage and the channel
pressure. With the general homogeneous solution for the acoustic channel pressure and the particular solution, which is the
pressure due to the actuator, in the frequency domain per frequency according to d’Alemberts solution:
2
P (z , t ) = Pr ei(ωt −kz ) + Pl ei(ωt +kz ) − αρ ceff
V
(134)
we can express the Paint current or signal in terms of propagating waves towards the nozzle and coming from the nozzle
with wave number k and frequency ω. Inserting this equation in Eq. (133) results in:
2
Ip = iω(Cp − α d33 Ap ρ ceff
)V + d33 bp
ω
k
Pl eiklp − Pr e−iklp .
(135)
Not only the drop formation, the refill and the wetting of the nozzle plate changes the reflection of the acoustic pressure
wave, but also an air bubble has a large influence on the pressure waves Pl coming from the nozzle.
6.2. Air entrainment
6.2.1. Wetting layer
Air entrainment is directly linked to the presence of an ink layer on the nozzle plate [352,279,485,419]. An ink layer
around the nozzle results in a lower drop speed, Section 5.4. The drop speed decreases with ink layer thickness up to 20 µm
and the nozzle ceases to fire drops with an ink layer thickness more than 20 µm. The experiment with adding a layer of
ink through the neighboring channels demonstrated this impact on the drop formation. The same experiment also shows
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H. Wijshoff / Physics Reports 491 (2010) 77–177
Fig. 85. Top view of an experiment where ink is supplied at t = 0 ms through the neighboring nozzles via a static pressure. The pictures are taken with a
high-speed camera with 10 kfps. The first frame shows the regular jetting. The second frame is taken at the moment when the ink layer reaches the jetting
nozzle. The third frame shows the moment when jetting stops. The last frame shows the air bubbles, which are created. These bubbles are inside the white
circle.
Source: Adapted from [279].
Fig. 86. (a) Flow3D simulation with an ink layer of 30 µm on the nozzle plate. (b–d) During the second actuation cycle, which started 50 µs after the first
cycle, an air bubble is entrapped.
Source: Taken from [279].
the formation of air bubbles. Four stages of this experiment are shown in a top view in Fig. 85a–d. First the normal jetting
situation is depicted. In the second frame the supply of an ink layer through neighboring channels is shown. The stopping
of the drop formation is shown in the third frame. Finally, the creation of air bubbles is depicted. After the drop formation
has failed, small air bubbles are visible in the ink layer as indicated.
The ink layer is applied during 1 s. The ink layer covers a large part of the nozzle plate and increases continuously in
thickness until it is very thick. The drop formation fails, but the actuation continues at a repetition rate of 20 kHz. Air bubbles
are visible in the ink layer after 0.2 s. After 1 s, the ink supply stops and the ink layer thickness decreases. After 1.5 s, jetting
starts again. It turns out that an air bubble is also entrapped in the jetting nozzle [279].
Flow3D simulations are used for a qualitative check whether air entrainment can be induced by the ink layer itself. With
an ink layer thickness of 0–20 µm, only the drop speed is lowered. With an ink layer thickness of more than 20 µm, no
drops are generated anymore. In a simulation with an ink layer thickness of 30 µm on the nozzle plate, air entrainment is
visible as shown in Fig. 86a–d. With this simulation it is assumed that the influence of the ink layer on the pressure waves
inside the printhead is negligible. Two actuation cycles at a repetition rate of 20 kHz are simulated. During the first cycle,
which started with no fluid movement at all, no air is entrained. Air is entrained during the second cycle. The actual air
entrainment occurs when the meniscus is pulling back and ink is already flowing into the nozzle. The void is then closed at
the top, because a lot of ink flows in from the layer around the nozzle. At an ink layer thickness of more than 40 µm, only a
net inflow of ink is visible without a large deflection of the free surface.
The experiment with an ink layer supplied through the neighboring channels shows the formation of multiple bubbles
in the ink layer. Because the ink layer thickness varies slowly on a seconds time scale, the critical ink layer is present during
multiple drop actuation cycles. Each cycle, air can be entrained. To determine the critical layer thickness, experiments are
performed with additional nickel plates to create a well defined ink layer thickness around the jetting nozzle [279]. Different
thicknesses of the additional nozzle plate are used to vary the ink layer thickness. By firing only three drops in one sequence,
the amount of ink around the nozzle is not changed significantly. The critical layer thickness turned out to be between 30 µm
and 40 µm, in agreement with the simulations. Below this critical layer thickness, the impact on drop formation is not strong
enough and above the drop formation process is hindered too much. Another source for the distortion of the drop formation
process are small dirt particles, which will be discussed in the next section.
6.2.2. Small dirt particles
With a wetted nozzle plate, a flow towards the nozzle is observed in most cases during jetting, Section 5.2.1. Relatively
small dirt particles, caught in the ink layer, are likely to reach the jetting nozzle and this can result in a distortion of the drop
formation. This distortion also influences the Paint signal. Therefore the acoustics inside the ink channels are continuously
monitored with the Paint signal [279]. At a repetition rate of 20 kHz, a time interval of only 30 µs between the actuation
pulses is taken for further analysis of the Paint signal. The amplitude of the current from the piezo element I (t ) during that
H. Wijshoff / Physics Reports 491 (2010) 77–177
157
Fig. 87. The amplitude of the Paint signal. During 25 drop formation cycles before and after a distortion the variance varies less than 0.5%, the distortion
leads to a deviation of 5%–10%.
Fig. 88. High-speed camera recording of the capture of the last jetted drops after a disturbance triggered the piezoelectric device. A piezo element with
a tiny metal chisel is visible at the left of the drop stream. After the trigger signal, the piezoelectric device moves into the stream of drops. The times are
given relative to the trigger moment. Four droplets, indicated with C, D, E and F, are caught. Jetting was stopped after droplet F.
time interval T is calculated:
σ2 =
1
T
Z
T
[I (t )]2 dt .
(136)
0
The amplitude of the Paint signal varies by 5%–10% during a distortion, Fig. 87. The amplitude of the Paint signal over 25
drop formation cycles before and after the distortion varies by less than 0.5%.
The influence of the distortion on the Paint signal is used to trigger a high-speed camera to record the drop formation
when the distortion takes place [486]. The correlation between the distorted drop formation and the variation in the Paint
signal is published in more detail in [279]. The deviation of the amplitude can not only trigger a high-speed camera but also a
piezoelectric device that can move 60 µm within 1 ms. This device, with a tiny metal chisel attached to it, was placed under
the nozzle plate at a distance of a few micrometers from the stream of jetted drops. When triggered, this device moves into
the stream of drops to capture them, see Fig. 88. At the same trigger signal also the actuation is stopped, and only the last
few drops are captured for further analysis. It turns out that the captured droplets contain relatively large particles with a
diameter of about 20 µm. These particles are not found in captured drops where no distortion occurred. The minimum in
the Paint signal always corresponds to the drop which is transporting the particle. At that moment, the nozzle is (partially)
blocked, which results in lower acoustic amplitudes inside the ink channel.
A variation of a few micrometers in the position of the meniscus results in a comparable variation of the Paint signal,
see also Section 3.4.1. To explain that the Paint amplitude is first several percent lower and then several percent larger, the
meniscus must first be protruded out of the nozzle and second the meniscus must be retracted. Also a variation in the drop
speed is observed during a distortion. The first droplet is 0.1–0.2 m/s faster and the second droplet up to 0.4 m/s slower,
with often a deviating shape and jetting angle [279]. The variation in the drop speed, as a result of the different meniscus
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H. Wijshoff / Physics Reports 491 (2010) 77–177
a
b
c
d
e
f
g
h
Fig. 89. Simulation of the effect of a spherical particle with a diameter of 20 µm on the drop formation. The pressure in the channel is kept the same and
the starting position of the meniscus is also at its original position. The drop formation is shown at 0, 15, 30 and 45 µs after start of the actuation. First with
the particle at 10 µm under the meniscus surface (a,b,c,d) and second with the particle at 30 µm under the surface (e,f,g,h).
Fig. 90. High-speed camera recording of the effect of a partial covering of the nozzle opening with a teflon wiper. Shown are the drop formation before,
during (the wiper moves away from the camera), and after the covering. The drop formation during the partial coverage is distorted a lot. After the distortion,
the drop formation is normal again.
positions (because of a variation in the refill level, Section 3.4.1), corresponds to the observed variations during the distortion
of the Paint signal.
The dirt particles itself can also be the source of the observed variations in drop speed. For a qualitative check, the drop
formation is simulated with dirt particles at different positions. In Fig. 89 two examples are shown with a spherical particle
at 10 µm and at 30 µm under the meniscus surface. The drop speed varies with the particle at different positions under the
surface. With respect to the undisturbed drop speed, the variations are:
•
•
•
•
•
•
at 5 µm, 1.9 m/s slower and the particle is jetted away
at 7.5 µm, 2.8 m/s slower and the particle is jetted away
at 10 µm, 1.3 m/s faster and the particle remains inside the nozzle
at 10 µm, 0.9 m/s faster and the particle remains inside the nozzle
at 30 µm, 0.4 m/s slower and the particle remains inside the nozzle
at 40 µm, 0.5 m/s faster and the particle remains inside the nozzle.
From the capturing of the drops, visually and with the piezoelectric element, we know that the last drop in the distortion
drags along the dirt particle. In the simulation this happens when the dirt particles are very close to the surface.
The variation of the amplitude of the Paint signal, shown in Fig. 87, is the most common example of a distortion. After
the disturbance occurred, jetting keeps going on as usual. But sometimes after a disturbance, an air bubble is entrapped,
which becomes later visible as a larger deviation of the Paint signal. Then, there is also a disturbance of the drop formation
visible, which finally even can lead to the complete breakdown of the drop formation process. This will be shown in the
next sections. The entrainment of an air bubble is always preceded by a distortion. No air bubbles are detected without a
distortion preceding it. So the described distortion is a necessary but not sufficient condition for air entrainment [279].
The exact mechanism of how air is entrained is still not known. In reality, the dirt particles have irregular shapes, mostly
in the shape of flakes. The drop formation model is a 2D model rotational symmetric model. To simulate irregular shapes a
full 3D model is required, which leads to a very long calculation time. The particles are relatively large. The particle size
is typically 20 µm, the nozzle diameter is 30 µm. Partial blocking of the nozzle opening could be the origin of the air
entrainment process. To check this, experiments are done with a 400 µm thick teflon wiper. During the jetting of a single
nozzle, the 1 mm wide wiper partially moves over the nozzle with a speed of 3 µm/µs. So the nozzle is partially blocked for
several actuation cycles. The drop formation is distorted a lot during this period, Fig. 90. If partial blocking of the nozzle is
the mechanism behind air entrainment, the experiment should lead to a deterministic generation of air bubbles. However,
this is not the case.
H. Wijshoff / Physics Reports 491 (2010) 77–177
159
Fig. 91. The change in bubble size (as normalized radius Rn = R(t )/R0 ) of bubbles with a diameter of 20 µm (black) and 40 µm (grey). The calculations
are done with the Rayleigh–Plesset equation and the pressure from the narrow channel model. The imposed pressure variation is large enough to generate
a collapse with an oscillation after the collapse [492].
This is also qualitatively confirmed with Flow3D simulations with a 3D drop formation model with a partially covered
nozzle opening. In all the simulations the drop formation is also distorted a lot, with a large deviation of the jetting directions,
but air entrainment is not seen. The asymmetric retraction of the meniscus does not give any clues on a larger chance to
entrain air.
6.3. The oscillating bubble
6.3.1. Size dynamics
The radius, position and size of an air bubble changes in an acoustic field [487–489]. For a bubble in an infinitely large
volume, the change of the bubble radius R of a bubble with radius R0 at ambient pressure is given by the Rayleigh–Plesset
equation [490]:
RR̈ +
3
2
Ṙ2 =
1
2γl
ρ
pg − p0 + pν −
R
−
4ηṘ
R
− p∞ (t ) .
(137)
The right term represents the forces generated through the pressure difference between the gas pressure inside the bubble
pg with respect to the ambient pressure, with corrections for the vapor pressure pν , the Laplace pressure, and the pressure
from viscous stresses. The last term, p∞ , is the imposed acoustic pressure at a distance where the pressure due to the inertia
of the radial flow field around the bubble is negligible with respect to the imposed pressure fluctuations. The polytropic
equation is used as the equation of state:
pV Γ = constant
(138)
with Γ the polytropic index, which is 1.0 for isothermal behavior and 1.4 for adiabatic behavior. For the gas pressure in the
Rayleigh–Plesset equation, the following expression can now be used:
pg =
p0 +
2γl
R0
− pν
R0
R
3Γ
.
(139)
The Rayleigh–Plesset equation is the basis for the explanation of the net displacements and the mass exchange of air bubbles
in an acoustic field.
It is assumed that the bubble oscillates isothermally, so that the polytropic index Γ = 1 [491]. The imposed pressure
fluctuation p∞ is calculated with the narrow channel model of the reference printhead, see Section 3.1. The resulting
dynamic changes in bubble size for bubbles with a radius of 10 µm and a 20 µm are shown in Fig. 91.
In Fig. 91 we can see that the bubble radius increases twice, at 9 µs and at 30 µs. This corresponds to the negative pressure
peaks in the pressure at the entrance of the nozzle, see also Fig. 32. When the imposed pressure becomes higher than the
ambient pressure, the bubbles collapses. After the collapse the bubble resonates with its natural frequency. The collapse and
resonance of a bubble occurs when the frequency of the imposed pressure field is much lower than the natural frequency
of the bubble [493]. The channel frequency of the 8 mm long pressure channel is 50 kHz and the natural frequencies of the
bubbles are 150 and 300 kHz for the 40 µm and the 20 µm bubbles, respectively.
The collapse of the bubble is an important effect and can generate many effects such as cavitation damage from the jet
formed with the asymmetric collapse near a solid wall [494,495] and can even excite sonoluminescence [496]. The jetting
effect can be used for a micro-pump [497,498] and the dynamic behavior of air bubbles is also used in an actuator [499]. In
our case, the oscillating bubble is a disturbing effect, which influences the channel acoustics and the drop formation.
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H. Wijshoff / Physics Reports 491 (2010) 77–177
Fig. 92. (a) The amplitude of the Paint signal when a distortion occurs before cycle 20 and air is entrained. The amplitude of the Paint signal deviates more
than 10 % within 100 drop formation cycles at a repetition rate of 20 kHz. The actuation amplitude is the nominal driving amplitude for a drop speed of
7 m/s. (b) The shape of the Paint signal without an air bubble (dotted line 1), with an air bubble (continuous line 2), and the difference between the two
signals (continuous line 3). This shows a 170 kHz distortion after 100 drop formation cycles.
6.3.2. Impact on channel acoustics
With acoustic transmitters and receivers in an acoustic pipe, several acoustic signals could be used for the exact detection
of air bubbles [500]. With our Paint measurement, only the global acoustic response can be measured. When air is entrained,
the amplitude of the Paint signal will show an increasing deviation after the distortion, instead of remaining at the same
value as shown in Fig. 87. At a repetition rate of 20 kHz, the variance will deviate more than 10% within 100 drop formation
cycles, Fig. 92a. It turns out that the deviation is caused by a typical pattern. This pattern becomes more clear when we look
at the difference between disturbed Paint signal and the signal without an entrained air bubble, the reference signal. When
looking at the shape of the Paint signal, Fig. 92b, it turns out that the deviation is caused by a distortion with a frequency of
170 kHz [279].
The question is what causes the 170 kHz distortion. It could be the afterbounce as shown in Fig. 91. After the collapse,
the bubble oscillates at its natural frequency. The natural frequency f0 of a free bubble, neglecting surface tension, viscosity,
vapor pressure and advection, is called the Minnaert frequency [482]:
f0M
=
1
2π R0
s
3Γ p0
ρ
.
(140)
The resonance frequency of 170 kHz corresponds to a bubble radius of 18 µm. However, the behavior of an air bubble in an
infinite space is different from the behavior in a bounded liquid [501,502]. Now the pressure field is not three dimensional
anymore. Between two plates it is two-dimensional [503] and in a long channel only one-dimensional [504,505]. The
confined space, where the bubble is located, has a large effect on the dynamic behavior of the bubble [506–508]. In our
case, the confined space is the semi-closed connection channel with a diameter of 250 µm and a small nozzle opening.
Within these constraints, the resonance frequency will be reduced by 30%–50% for a bubble with a radius of 10–20 µm [509,
510]. So the radius of the bubble causing a 170 kHz resonance is only 9–12 µm. The volume of such an air bubble is 2–8 pl,
much smaller than the volume displacement by the acoustic pressure field.
The effect of the confined space can be taken into account by adding the pressure disturbance caused by an air bubble
to the imposed pressure in the Rayleigh–Plesset equation. When a bubble shows a collapse in the response to the imposed
pressure field, also thermal and acoustic damping should be taken into account [511,512] and in extreme cases even the
minimum radius, which is determined by the excluded volume of the gas molecules [513]. This makes the description of the
dynamical behavior of a bubble in an acoustic field rather complicated and is not taken into account in our analysis.
The imposed pressure is calculated with the narrow channel model. With an air bubble with a volume Vb between the
channel and the nozzle, the continuity equation can be written as [485]:
dVb
= 4π R2 Ṙ = An un − Ach uch .
(141)
dt
With this equation added to the narrow channel equation and the Rayleigh–Plesset equation, the effect of the confined
space can be calculated. The change of the bubble radius in the reference printhead driven at normal actuation amplitude as
calculated with and without the effect of the confined space is shown in Fig. 93. The confined space dampens the collapse
and afterbounce almost completely.
With regular printheads, we can only measure the distortion of the drop formation and the impact on channel acoustics
through Paint. A direct measurement of air bubbles is only possible with an optical detection in special, transparent test
printheads [483], see also Fig. 95. A transparent nozzle plate and a transparent connection channel are made with powderblasting out of glass plates. The transparent connection channel has a length of 400 µm and a hourglass shaped cross-section
H. Wijshoff / Physics Reports 491 (2010) 77–177
a
b
25
RP
Conf_space
28
26
20
24
15
R /µm
Bubble radius [µm]
161
22
20
10
18
5
16
Time [µ s]
0
0
5
10
15
20
14
0
10
20
30
40
50
t /µs
Fig. 93. (a) The calculated change of the radius of a bubble in a reference printhead at normal driving amplitude as calculated with the Rayleigh–Plesset
equation without and with the effect of the limited space around the air bubble. (b) The measured (dots) and calculated (line) change of the bubble radius
in a special transparent head, with comparable acoustic properties.
Source: (b) taken from [514].
with a diameter between 80 µm and 250 µm. The 70 µm long conical shaped nozzle has an entrance diameter of 50 µm
and an outlet diameter of 30 µm. These two glass plates replace the nickel nozzle plate. The pressure channel of this head
has a length of 5 mm and there is also the normal connection channel with a length of 1 mm. The acoustic properties of this
printhead are still comparable with the opaque reference head. High speed camera recordings with a Phantom V7 camera
at 40 kfps and a continuous light source behind the glass plates show the relation between the behavior of an entrained air
bubble and the impact on channel acoustics and the drop formation.
Also in the optical recordings, there is no afterbounce visible, Fig. 93b and the measured data fits well with the calculated
results. So the resonance at the natural bubble frequency after a collapse is not the cause of the 170 kHz distortion in the
Paint signal.
The distortion is caused by the fact that the acoustic properties of the channel are changed. The reflection of the pressure
wave will be altered by the presence of an air bubble as we will see in Section 6.5.3. We see the influence of a bubble as
a 170 kHz distortion because this frequency is next to the basic channel resonance at 50 kHz the next strong peak in the
frequency characteristics of this head. The frequency characteristics describe the relation between the electric activity in the
actuator and the meniscus movement and acts also as a filter for the Paint measurements. The high frequency components
are more sensitive to small changes in the reflection conditions of the pressure waves and therefore, the higher frequency
components are altered more than the low frequency components.
6.3.3. Impact on drop formation
In the experiments with the transparent head, the oscillations of a bubble have an effect on the drop speed [485]. This
is shown in Fig. 94a for a range of bubble radii between 11 and 19 µm. The driving amplitude is low, which results in a
drop speed of only 1 m/s without an air bubble. In the experiment, the droplet velocity increases and reaches a maximum
of 2.5 m/s. Then the droplet velocity gradually decreases to 1 m/s at 0.9 s. A small amplitude oscillation with f = 50 Hz is
superimposed onto the droplet velocity, reflecting the AC frequency of the devices. This effect is negligible compared to the
effect of the entrained air bubble on the drop speed.
The recorded images of the ejected droplets and oscillating bubbles are analyzed with a gray-level threshold to determine
the location of the edges of the droplets and bubbles. The images of the bubbles only consist of some tens of pixels, limiting
the accuracy of the size determination. Other sources of errors are the optical diffraction and the assumed sphericity of the
bubbles in the digital image analysis. For the ejected drops and for the bubbles within the channel (away from the walls) we
do not have any indication of deviations from sphericity. However, the air bubbles, when pushed against the glass nozzle
plate, seem to be slightly non-spherical.
The scatter in the bubble radius is found to be quite large, due to the low contrast in the images and because multiple
bubble radii are measured over one acoustic cycle, see also Fig. 93b. When we compensate for the latter by plotting the
average radius (over 40 bubble radii), the bubble growth becomes more obvious. This is used in Fig. 94b where the two data
sets are combined to get the drop speed as a function of bubble size. The measured drop speed increases to 2.5 m/s with
increasing bubble size until a radius of 16.5 µm and than decreases very fast with increasing bubble size. Finally, jetting fails
when the bubble radius becomes larger than 19 µm.
A resonance effect of the bubble oscillation with the meniscus movement results in the increase of the velocity. This can
be verified with a simple model for a harmonic oscillator [515], which is based on the approach used in [509]. The dynamic
behavior of the bubble is based on the polytropic relation pV Γ = constant between bubble volume and the pressure. The
meniscus movement is based on Newton’s second law. The viscous dissipation is taken into account with a Poiseuille flow
profile in the nozzle to correct for the pressure loss between the pressure at the bubble and the ambient pressure. The
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H. Wijshoff / Physics Reports 491 (2010) 77–177
b
18
udrop
2
16
1.5
14
1
12
0.5
10
0
0.2
0.4
0.6
t /s
0.8
3
2.5
]
20
–1
2.5
[m s
22
udrop
3
R bubble / µ m
–
/m s 1
a
2
1.5
1
0.5
1
0
0
5
10
15
20
25
30
bubble radius [ µ m]
Fig. 94. (a) Simultaneously measured bubble radius and drop speed with the transparent head described in [485]. The actuation is started at t = 0 s. The
bubble diameter increases from 22 to 38 µm and the drop speed increases from 1 to 2.5 m/s at 0.2 s and decreases again. (b) The resulting relationship
between bubble diameter and drop speed as measured (solid line) and calculated with the simple harmonic oscillator model (dashed line). The measured
drop speed shows a maximum at a bubble diameter of 33 µm. Jetting fails with a bubble diameter larger than 38 µm.
continuity equation completes the set of equations. After linearization around the equilibrium volume, a forced harmonic
oscillator equation for the meniscus speed is obtained. A sinusoidal velocity with frequency f is imposed with the main
channel resonance frequency, which is 50 kHz for the transparent head. The radius of the bubble where the resonance effect
takes place reads according to this model [485]:
Rres
0
3Γ ρ A3n p0
=
16π 3 ln (ρ 2 f 2 A2n + 16η2 )
31
.
(142)
In the model, the conical nozzle shape is replaced by a straight nozzle with the same outlet diameter as the conical shaped
nozzle, i.e. 30 µm. The conical nozzle has an entrance diameter of 50 µm and a length of 70 µm. This is transformed to
a straight nozzle with the same acoustic impedance, see Section 3.1.2 and Eq. (66). The length of the acoustic equivalent
straight nozzle is 32 µm. The resulting drop speed is shown in Fig. 94b, together with the measured results. The drop speed
is calculated with the energy balance as mentioned in Section 3.2.2. The results of the simple model agree reasonable with
the experimental data. Only the bubble radius where the resonance effect takes place is a little smaller, 13 µm instead of
16.5 µm.
The eigenfrequency of a bubble with radius R0 which acts as a harmonic oscillator with the meniscus movement of a
nozzle is:
s
f0H
=
3Γ R2n
4π ρ ln R30
.
(143)
The eigenfrequency of a bubble with radius D0 = 16.5 µm is 50 kHz, the same as the main channel resonance frequency.
6.4. The moving bubble
6.4.1. Balance of forces
Many forces are acting on a bubble in an acoustic pressure field. The movement of a bubble is determined by the
competition between the acoustic and the hydrodynamic forces [516]. First there is the primary Bjerknes force. The force
acts in the direction of the acoustic pressure gradient. A small bubble will be pushed towards the pressure node, i.e. the small
bubble at the nozzle will move into the channel. A very large bubble will have a low natural resonance frequency and start
to oscillate in anti-phase with the acoustic pressure field [517]. The primary Bjerknes will now act in the opposite direction
towards the pressure anti-node, i.e. the bubble will move towards the nozzle plate.
The secondary Bjerknes forces are generated by the acoustic pressure field emitted by the oscillating bubble itself or by
another bubble. The basis equation is the same as the equation for the primary Bjerknes force, but now the pressure field
is generated by the bubble itself. Due to a mirror effect, this force will push bubbles against a rigid wall. The same mirror
effect results in a repulsing force against a free surface, e.g. the meniscus [518]. Many expressions can be obtained for the
secondary Bjerknes force.
The two Bjerknes forces can explain the main displacement characteristics of the only directly experimentally observed
displacement of a bubble. This is the optical measurements with the transparent printhead, Fig. 95 [484]. A small bubble,
which is generated near the nozzle, moves with a speed of 5 mm/s into the channel, because of the primary Bjerknes force,
and is after 118 ms attracted against the (transparent) connection channel wall, because of the secondary Bjerknes force.
H. Wijshoff / Physics Reports 491 (2010) 77–177
163
Fig. 95. (a) The transparent head, used for direct visual measurements of entrapped air bubbles with the nozzle plate 1, the connection channel 2, an air
bubble 3, and a neighboring channel 4. (b) The y position of the air bubble as a function of the time after the air entrapment. The bubble first moves into
the channel. At t = 52 ms the Bjerknes force reverses sign and the air bubble is pushed back towards the nozzle.
Source: Taken from [484].
After a while this bubble has grown in size to a diameter of 32 µm. The natural frequency of this bubble as harmonic
oscillator, as described in the previous section, is reduced to 55 kHz and tends to become lower than the frequency of the
main channel resonance. The bubble moves after 55 ms in the opposite direction towards the nozzle plate with a speed of
20 mm/s.
The drag force from the ink flow becomes important especially near the nozzle itself [519–521]. This force tends to drag
the bubble along with the ink flow, thus along with the drop formation. Buoyancy or gravity forces play no role in our case.
Other forces which could play a role are lift forces from the radial flow field [522,523], the added mass and inertia forces and
finally the Basset history force which results from the absorbed momentum in the boundary layer around the bubble [524].
These acoustic and hydrodynamic forces will result in complicated movement patterns, which will be discussed in the next
section.
6.4.2. Net displacements
The forces on the bubble result in a displacement pattern, which can only be resolved numerically for an opaque
printhead. In Fig. 96a the net displacement after one drop formation cycle is shown. For a bubble with a diameter of 10 µm
at different positions on the nozzle axis. The results are simulated with a 2D rotational symmetric drop formation model in
Flow3D. So only bubbles at the nozzle axis can be simulated. The pressure inside the bubble is calculated with the polytropic
equation. The initial pressure inside the bubble is equal to the ambient pressure, enlarged with the Laplace pressure 2γ /R0 .
The pressure in the connection channel at a position 300 µm in the channel in front of the nozzle is kept the same. There is
only a one-way coupling between the channel acoustics and the flow phenomena in the nozzle. The bubble cannot influence
the pressure waves in the channels in these simulations.
Depending on the axial position, the bubble is either moved inwards or outwards after one drop formation cycle. There
are four positions where the net axial displacement becomes zero. Two positions are unstable because the bubble at locations
around this point moves further away from these positions. Those are the position at 25 µm and 72.5 µm. But, the two axial
positions at 62.5 µm and 105 µm are stable. The bubble at locations around points moves towards these positions, so these
two positions can act as a bubble trap.
At axial positions between the meniscus and 20 µm behind the surface, the bubble is jetted out. Also experiments show
that jetting out of an air bubble is a possible recovery mechanism. In Fig. 96b the measured amplitude of the Paint signal is
shown. At drop formation cycle 70, a distortion occurs followed by air entrainment. At drop formation cycle 124, a second
distortion is visible without any consequences. With drop formation cycle 155, the amplitude of the Paint signal returns in
one cycle to its normal value, which can only be caused by jetting out of the bubble. The numerical simulation of the jetting
out is shown in Fig. 97. The bubble moves along with the drop formation. The drop shape will be distorted, but after this
drop formation cycle, no deviations will be found anymore.
Numerical simulations of complete movement patterns with Flow3D require a huge computational effort. For many
starting positions, the net displacement has to be calculated over many drop formation cycles. Furthermore, in most cases
off-axis positions have to be simulated to get a complete picture. This requires the use of a full 3D drop formation model in
Flow3D. However, modeling off-axis bubbles results in extreme long CPU times with Flow3D because of the very fine grid
size needed over a large area. An alternative numerical option is to use a boundary element method for the simulations of
bubble dynamics [525–528] or to use a lumped parameter approach [529], where the equations for the individual forces are
added to a Navier–Stokes solver for the flow in the nozzle.
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H. Wijshoff / Physics Reports 491 (2010) 77–177
a
b
Bubble Net Transport [um]
0.13
0.12
σ2
0.11
0.10
0.09
60
80
100
120
140
160
droplet number
Fig. 96. (a) Simulation of the net bubble displacement after one drop formation cycle. Depending on the axial position the bubble with a diameter of 10 µm
is moved outward or inward. At axial positions around 62.5 µm and 105 µm the bubble can be trapped. At axial positions between 0 µm and 20 µm the
bubble is jetted out. (b) Measured amplitude of the Paint signal with distortions at drop formation cycles 70 and 124, air entrainment after drop formation
cycle 70. The jetting out of a bubble is visible at drop formation cycle 155. This is a fast recovery mechanism, generated by the net displacement pattern
inside the nozzle.
Fig. 97. Simulation with Flow3D of the jetting out of a bubble. The bubble has a diameter of 10 µm at an axial position 20 µm under the meniscus. The
frames show the drop formation with the bubble at 0, 8, 16 . . . 56 µs after start of the actuation. The channel length is 8 mm and nozzle diameter is 32 µm.
6.5. The growing bubble
6.5.1. Rectified diffusion and dissolution
A bubble in an acoustic pressure field will grow by rectified diffusion when the acoustic pressure variations are strong
enough [490,491,530,531]. At pressure maxima, air is squeezed out of the bubble, but this loss is overcompensated at the
pressure minima when the bubble expands. This results in a net gas diffusion into the bubble. Rectified diffusion is a result
of a surface effect and a shell effect, e.g. an expanded bubble can absorb more air because of its larger surface area and the
higher concentration gradient of the dissolved air in the liquid around the bubble, which is compressed by the expanded
bubble.
In the transparent head, the growth rates are optically measured. The measured net growth rate is 0.04 pl/ms when
actuating at 10 kHz with a relative low driving amplitude for a drop speed of 1 m/s [485]. This is a net growth of
0.004 pl/cycle. With a very low non-jetting actuation amplitude at a repetition rate of 20 kHz, the net growth rate turns out
to be much larger, 0.39 pl/ms or 0.02 pl/cycle [484]. This difference is an indication of an influence of the flow field around
the bubble.
Just after the air entrainment, the bubble has a radius of 6 µm (0.9 pl) as measured visually with the transparent heads
[484]. In the opaque reference printheads, the bubble size is estimated at 20 µm (4 pl) after 100 drop formation cycles at
a repetition rate of 20 kHz with a nominal driving amplitude for a drop speed of 7 m/s, Section 6.3.2. With the assumption
that the bubble size just after the air entrainment is the same as in the transparent heads, we get an estimation of the net
growth rate of 0.62 pl/ms or 0.03 pl/cycle.
Without an acoustic pressure field, the bubble will dissolve. There is no net transport of ink and the ink around the bubble
will saturate with dissolved air and the dissolution rate will be rather low. In the experiments without actuation with the
transparent head [485], the dissolution rate of a 26 µm bubble is constant at 0.5 pl/s until the bubble is completely dissolved.
This is shown in Fig. 98. The dissolution rate is 2 to 3 orders of magnitude slower than the growth by rectified diffusion. This
can also be seen in the figure, where the bubble size increases much faster when the actuation is started.
H. Wijshoff / Physics Reports 491 (2010) 77–177
a
165
b
80
60
second equilibrium
volume [pl]
volume [pl]
60
40
20
20
first equilibrium
jetting
starts
0
0
50
100
40
150
time [s]
200
0
250
0
10
20
30
40
time [s]
Fig. 98. Measured bubble size with actuation stopped at t = 0 s until t = 165 s. The dissolution rate is 0.46 pl/s. Then the actuator is turned on and
droplets are jetted from the nozzle. The bubble size increases fast and saturates at an equilibrium size of 20 pl within a second. The equilibrium is broken
after 12 s at the moment jetting breaks down. This is shown in the second figure, which shows only the first 40 s after the actuator is turned on again.
Without drop formation the bubble size saturates at a much larger size of almost 70 pl.
The bubble size increases fast and saturates at an equilibrium size of 20 pl within a second. After 1000–5000 actuation
cycles, the growth by rectified diffusion is compensated by the dissolution of the bubble. This equilibrium is broken at the
moment when jetting breaks down. Without drop formation the bubble size saturates at a much larger size of almost 70 pl.
With jetting, the ink around the bubble will be continuously refreshed. The ink around the air bubble will be less saturated
with dissolved air. This results in higher dissolution rates. The saturation size in the jetting situations will be smaller than in
the non-jetting situations. Experimental data are only obtained with a transparent head. In the jetting situations the bubble
size saturates always at a radius of 17–20 µm (20–33 pl) and in the non-jetting situations at a radius of 22–25 µm (50–70
pl) [484,485,483].
For very large bubbles, viscous forces and the surface tension forces are not strong enough anymore to limit the growth
of bubble shape instabilities [492,532,533]. Shape instabilities will also limit the growth of bubbles, but are not likely to
occur in the confined space of an ink channel and the observed bubbles sizes so far.
6.5.2. Influence of the actuation
When turning off the actuation for a specific time interval, the bubble will either be entirely dissolved or only partly.
This can be seen in the Paint signal after the actuation is started again. If the time interval is long enough, the bubble will be
dissolved completely, and the Paint signal is normal again. When the time interval is not long enough, the bubble is dissolved
only partly. The bubble is still present and the modification of the Paint signal by the bubble will be still visible. The bubble
will again grow by rectified diffusion until it saturates at its equilibrium size.
This is in fact what happens in the experiment with the regular head, where the growing and dissolution cannot be
measured optically. A bubble is created during an actuation at a high drop repetition rate. Then the repetition rate is lowered
stepwise and the Paint signal is monitored until a new equilibrium is established. By lowering the repetition rate the bubble
will get more and more time to dissolve until it is dissolved completely before the next drop is jetted. The results of this
experiment are shown in Fig. 99.
The analysis of the Paint signal is this figure is as follows. The current from the piezo element I (t ) during a time interval
T is compared with a reference I0 (t ) (i.e. the signal without air bubbles) and the variance σ 2 of the difference is calculated:
σ2 =
1
T
Z
T
[I (t ) − I0 (t )]2 dt .
(144)
0
At a low actuation amplitude, the rectified diffusion is slow and the bubble dissolves completely in a relatively short time.
Lowering the repetition rate to 1–7 kHz is already sufficient at actuation amplitudes of 50%–80% of the nominal amplitude,
which is 35 V for this printhead. At nominal actuation amplitude the repetition rate must be lowered to 200 Hz before the
bubble dissolves completely.
Because of jetting out along with the drop, bubbles can disappear already at 400 Hz at the nominal driving amplitude. This
is shown in Fig. 99b+c, where the variance of the difference between the Paint signal and the reference signal is depicted as
a function of the decreasing repetition rate. In Fig. 99b, the decreasing deviation of the Paint signal is shown when a bubble
dissolves completely between a repetition rate of 6 and 3.5 kHz. In the lower picture the decreasing deviation is shown
when the bubble is jetted out at a repetition rate 285 Hz. The jetting out of the bubble is indicated by the stepwise decrease
of the Paint signal to the normal value.
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H. Wijshoff / Physics Reports 491 (2010) 77–177
Fig. 99. (a) Measured drop repetition rate at which the air bubble, which is created at a repetition rate of 20 kHz, dissolves completely when lowering
the repetition rate. At low driving amplitude the rectified diffusion is weak and the bubble dissolves already at a high repetition rate. At a high actuation
amplitude, the repetition rate must be lower before the bubble disappears. (b) The decrease of the deviation of the Paint signal (as variance of the difference
with the reference Paint signal). At 22 V the bubble dissolves and the deviation decreases gradually to zero (upper picture). (c) At 35 V the bubble can be
jetted out and the deviation decreases stepwise to zero (lower picture).
Fig. 100. Measured (a) and calculated (b) effect of a large air bubble on the Paint signal. A large air bubble acts as an open boundary condition for the
pressure waves inside the channel. The acoustic properties of the channel shift to a 21 λ resonator with a frequency of 65 kHz instead of 50 kHz. The frequency
and also the amplitude of the acoustic pressure increase.
6.5.3. Impact on the actuation
At high repetition rates of 10–20 kHz and nominal actuation amplitudes, air bubbles grow during 1000–5000 actuation
cycles to a bubble size of 50–70 pl. This volume is comparable to the total volume displacement by the acoustic pressure
wave. The volume displacement of acoustic pressure field can be counteracted by the change in bubble size. This will change
the acoustic properties of the channel completely.
Without an air bubble, the nozzle acts as a partial closed boundary condition for the pressure waves inside the ink channel,
see Section 3.1.2. The channel will act almost as a 14 λ resonator. A large bubble can counteract the pressure build-up. This
results in a complete open reflection. The channel will act now as a 12 λ resonator. The main resonance frequency will increase
and also the amplitude of the pressure wave will increase. This is visible in the measured and in the calculated Paint signal,
Fig. 100. The Paint signal is calculated with Flow3D and shows the same general trend. The simulations with the acoustic
model can only show the effect of air bubbles with diameters of more then 30 µm, because the grid size in the nozzle is
3–4 µm.
To simulate the effect of a wide range of bubble sizes on the pressure waves inside an ink channel, the narrow-channel
model is extended [514]. To keep the calculation time as low as possible, only cylindrical geometries are considered. The big
advantage is that for the velocity profile in the frequency domain, see equation 3.22 for a rectangular cross-section, which
has an open form, a much more efficient closed form equation can be used [534]:
Bch = 1 − 2
J1 (kRch )
kRch J0 (kRch )
(145)
H. Wijshoff / Physics Reports 491 (2010) 77–177
167
Time [s]
Fig. 101. (a) The measured and calculated Paint signal as difference between the disturbed and undisturbed signal from the new transparent head,
measured after an actuation with a 454 driving waveform with an amplitude for a drop speed of 6 m/s with a bubble volume of 81 pl and 86 pl respectively.
(b) The direct optically measured bubble volumes (blue) and the bubble volumes derived with the model from the acoustic Paint signal (dotted line). At
small bubble volume, the difference becomes rather large, as shown in the insert.
Source: Taken from [535].
with Jn the nth order Bessel function. A linear form of the Rayleigh–Plesset equation without acoustic and thermal damping
in the frequency domain with only harmonic perturbations around the radius R0 at ambient pressure is used:
− ω2 r =
1
ρ R0
4η
2γl
3Γ
2γl
−r p 0 +
− pv
+ r 2 − iωr
− p∞
R0
R0
R0
R0
(146)
with p∞ the pressure at a large distance. Together with the continuity Eq. (141), the impact of an air bubble on the channel
acoustics can be calculated accurately.
With a second printhead setup with a transparent section, but now with a regular electroformed 100 µm nickel nozzle
plate, the dissolution of an air bubble is recorded optically, and during this process, the Paint signal is recorded by actuating
and listening with a repetition rate of only 1 Hz [535]. This means there is practically no influence of the actuation on the
dissolution of the air bubble. The next step was to match the measured Paint signals with calculated Paint signals. In Fig. 101a,
an example of a measured and a calculated Paint signal is shown.
With the corresponding calculated Paint signal, the bubble size can also be determined theoretically. The result is shown
in Fig. 101b. The acoustically derived bubble volumes deviate less than 12% from the directly optically measured bubble
volumes, except for very small bubble volume, where the non-linear behavior of an air bubble starts to play a role [529].
When a distortion leads to air entrainment, the bubble radius is 10 µm after hundred drop formation cycles. The drop
speed remains the same during the first 100 drop formation cycles in reference printhead. The bubble continues to grow
and the drop speed starts to decrease with 1 m/s during the next 300 drop formation cycles [279].
In the experiments with the transparent head [485], jetting breaks down at a bubble radius of 19 µm. The bubble size
is then of the same order as the volume displacement by the acoustic pressure field. The bubble counteracts the acoustic
pressure variation. There is no driving force left for the ink movement in the nozzle and the drop formation comes to an end,
as validated with acoustic simulations in Flow3D [130].
Bubbles can also disturb the drop formation directly. During the meniscus movement, the capillary pressure changes
because of the changing curvature of the meniscus. The decreasing capillary pressure when a drop starts to neck results in a
larger bubble size and thus in an increasing flow rate back into the nozzle with more increasing necking and more decreasing
capillary pressure etc. [480]. However, in our fast drop formation process, the capillary pressure is too small with respect to
the total pressures involved to see any influence of this effect.
6.6. Concluding remarks
In Section 5 the wetting of the nozzle plate was discussed. The wetting of the nozzle plate is also a source for jetting
instability, the subject of this section. There is a critical layer thickness, which directly leads to air entrainment. The
generation of air bubbles is the mechanism, which can lead to nozzle failure. The wetting layer plays also a role in
combination with 20 µm small dirt particles, caught in the ink layer and transported towards the nozzle. The distortion
by a dirt particle can be detected with an acoustic measurement, which uses the piezo elements also as sensors.
An entrapped air bubble will oscillate in the acoustic pressure field. The acoustic properties of the ink channel act as a
filter for this measurement. Therefore the influence of small air bubbles is recorded as a signal with a frequency of 170 kHz.
Primary and secondary Bjerknes forces, drag forces and many other forces result in movement patterns of the air bubble. The
movement of an air bubble can result in jetting out, thus in a fast recovery. The movement of air bubbles can be visualized
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H. Wijshoff / Physics Reports 491 (2010) 77–177
with a special transparent head. Small air bubbles do not have a negative effect on the drop formation process. However,
in the pressure field the bubbles will grow by rectified diffusion. The growth rate is much faster than the dissolution rate,
the other recovery mechanism. The air bubbles grow in the pressure field to a size comparable with the drop size and the
displacement by the acoustic pressure waves. The large air bubble will then counteract the pressure build-up and the drop
formation process ceases completely.
7. Conclusions and outlook
In the past decades, inkjet technology has evolved into a technology which plays an important role in the graphical
printing industry and in many emerging new industrial and medical applications. The piezo inkjet technology has unique
capabilities due to its ability to deposit a wide variety of materials on various substrates in well defined patterns. To
comply with the increasing and diverging requirements for today’s inkjet technology, a fundamental understanding of the
underlying processes is very important.
The physics behind the chain of processes comprise the two-way coupling from the electrical to the mechanical domain
through the piezoelectric actuator, the coupling to the acoustic domain inside the ink channels, and the coupling to the
fluid dynamic domain, the drop formation process. Furthermore, wetting of the nozzle plate and air bubbles can have a
negative influence on the printhead performance. The five topics (actuation, channel acoustics, drop formation, wetting,
and air bubbles) are the sections of this article.
The first step is the actuation, which implies the transformation of an electric voltage to a deformation of the ink channels.
With a piezoelectric actuator design, an efficient transformation of electric potential to mechanical deformations can be
realized. However, mechanical constraints can result in cross-talk effects, and resonances in the printhead structure can
be excited. The deformations inside the printhead cannot be measured. Modeling with a commercial finite element code
is a way to resolve this. The numerical modeling provides the design rules for an efficient actuation, without generating
cross-talk effects. The drive towards smaller printheads with more nozzles will push the balance between the operational
demands and the manufacturability to its limits. A printhead design is a compromise between the two aspects, the functional
requirements and the cost of production. The numerical modeling is the tool to match the often conflicting requirements
for the different parts in the printhead. This leads to an optimal printhead design within known constraints. New actuation
principles can also be explored, before starting the expensive and time consuming manufacture of a new printhead design.
The deformation of the channel cross-section results in pressure waves. The pressure waves in the ink channels play a
central role in the printhead operation, Section 3. They connect the electrical–mechanical domain of the actuator to the fluid
dynamic domain in and outside of the nozzles. The traveling wave provides an efficient acoustic mechanism to generate the
required pressure for the drop formation process, when the acoustic impedance of the nozzle matches the impedance of the
ink channel. Also refill, the filling of the nozzle for the next drop formation cycle, is driven by the acoustic pressure waves.
However, residual vibrations in the ink channel must be as low as possible before the next drop formation cycle starts.
Another disturbing effect is acoustic cross-talk, which is the traveling of pressure waves through the reservoir towards
neighboring channels. This can be avoided with a thin foil on the ink supply. Measurements are done with the laser-Doppler
setup and an acoustic measurement, which uses the piezo elements also as sensors. Details of the phenomena inside the
ink channels, such as local pressure variations, are available through modeling. The acousto-elastic interaction plays an
important role in the ‘‘narrow-channel’’ model and acoustic modeling with Ansys provides the details of the interactions
with the structure dynamics of the printhead. With fundamental knowledge of the acoustic properties, new printhead
concepts can be developed successfully into an operating printhead that can fire billions of drops with constant size and
velocity and a reliability as high as possible. The flow and pressure at the nozzle sets the boundary condition in the acoustic
modeling. The pressure at the entrance of the nozzle is the coupling to the final stage of the printhead operation, the drop
formation.
The application of the traveling wave principle leads to the drop formation process, which takes several tens of
microseconds. The drop formation process is the subject of Section 4. Optical measurements and modeling provide the
information, which leads to its better understanding. The tail break-off process is not affected by the actuation. Up to
four satellite drop formation mechanisms can generate extra drops, which have a negative impact on the printing result.
An important application is drop size modulation. With the knowledge of the drop formation process, eight drop size
modulation techniques are identified. For the modeling of the free surface flow with surface tension and its impact on
channel acoustics we use the commercial volume of fluid code Flow3D. The calculations provide also information on the
meniscus retraction, the internal distribution of the speed of the ink, and the impact of the drop formation process on
the pressure waves inside the ink channels. The Flow3D software is adapted to incorporate flexible walls, which was until
recently not feasible in the standard code. We can predict the resulting drop properties accurately. This will be used to
investigate the jetting behavior of many new liquids, which are used in different new applications. The critical material
properties can be defined and this will be used as a guide for the development of new jetting materials. The jetting of
smaller drops is more sensitive to the influence of the surrounding air. This is a new challenge for the modeling of the drop
formation, requiring the development of new models.
The flow of ink on the nozzle plate is important. Wetting of the nozzle plate can influence the drop formation and the
pressure waves inside the ink channels. The material interactions, which determine the wetting properties, are not known
in detail for many materials. Therefore, an experimental study of the wetting phenomena is performed with a nickel nozzle
H. Wijshoff / Physics Reports 491 (2010) 77–177
169
plate, Section 5. An actuated nozzle induces several flow patterns in an ink layer around the nozzle opening. When jetting,
an induced air flow also influences the flow pattern around a nozzle. The nickel nozzle plate and the ink are in the complete
wetting regime, except for a central band of (older) ink in the middle of the nozzle plate, which is in the partial wetting
regime. The precursor film provides a means of communication between the central band of ink and the nozzles. Differences
in surface tension of the ink result in Marangoni flow over the entire nozzle plate. The experimental observations can be
understood with theories described in the literature. A numerical model, which can predict the wetting properties, is not
available. The molecular interactions have to be taken into account. However, molecular modeling will only provide the
interaction parameters. The parameters must be translated into a macroscopic flow pattern with additional models. This is
closely related to the research on the interactions of ink with the media. The impact, spreading, and solidification of drops
onto a solid surface is the final stage of the printing process. New applications require the depositions of various kinds of
liquids on different types of substrates. Numerical modeling of the wetting phenomena, which can take into account the
material interactions, would be a very powerful tool in the development of new inkjet applications. The final goal is to
control the wetting properties of the nozzle plate. Either a uniform thin layer of ink or no ink layer at all will improve the
jetting behavior of the printhead.
Wetting, without or with dirt particles, can also result in air entrainment. Air bubbles play an important role in the jetting
stability. The theoretical and experimental research on the generation and the behavior of air bubbles and their impact on the
printhead performance is the subject of Section 6. The resulting drop properties, the Paint measurement and visualization
with a special transparent head are the experimental tools. Various aspects of the generation and the behavior of air bubbles
are clarified. An air bubble oscillates in the acoustic pressure field. The oscillations can result in an enhanced drop speed and
also the acoustic pressure waves are altered, as measured with Paint. Many forces act on an air bubble, which results in
complicated movement patterns. The air bubble grows in the acoustic pressure field by rectified diffusion, with a growth
rate much faster than the dissolution rate. Large air bubbles disturb the drop formation. The ultimate goal is to prevent the
generation of air bubbles, or to control the behavior of air bubbles. Small air bubbles, which do not have a negative influence
on the drop formation, can be manipulated with acoustic pressure waves. Either a controlled jetting out or a fast dissolution
would remove an air bubble before the drop formation is disturbed. The air bubble also influences the acoustic properties
of the ink channels. Therefore, a numerical model will have to incorporate a full two-way coupling between the dynamic
behavior of an air bubble and the pressure waves inside the ink channels. Only with numerical modeling, the results of the
acoustic measurements can be translated into bubble properties such as bubble radius, position, velocity, etc.
The requirements for the inkjet printing technology will increase. The near future demands for smaller drops, higher drop
velocities, higher drop repetition rates, smaller printhead designs, a maximum jetting stability, complex fluids, etc. To meet
these challenges, inkjet research will continue towards high integration densities in the printhead, zero cross-talk levels,
zero variation of the resulting drop properties, and the elimination of nozzle failure. Also many new jetting materials will be
explored, even the jetting of molten pure metals is one of the new applications. Material interactions will play an important
role, which pushes the modeling to bridge the gap between continuum theories of our macro world and the molecular,
particle based, theories of the micro- and nano-world. This makes the development of new inkjet technologies an exciting
environment, where scientific research and industrial development can reinforce each other for many years to come.
Acknowledgements
The research, described in this thesis, is done at the R&D Department of Océ Technologies B.V., in close cooperation with
the Physics of Fluids research group of the University of Twente. I want to thank all the people who have contributed to this
research, especially my Océ colleagues, Hans Reinten, Marc van den Berg, and Wim de Zeeuw and from the University of
Twente especially Prof.Dr. Detlef Lohse and PhD students Jos de Jong, Roger Jeurissen and Arjan van der Bos.
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