Bubble Nucleation Pulse-Width in Thermal Inkjet Actuators Open Access

Thermal inkjet technology is used in traditional printing applications such as documents, labels, and packaging, as well as additive manufacturing, electronics, and pharmaceutical manufacturing [1–3]. The underlying principle is the delivery of heat to the ink via Joule heating of a conductive element located adjacent to the ink. A bubble forms, expelling the liquid through a nozzle, which forms a droplet that traverses the print gap to land on the target surface.

Thermal inkjet actuators are typically multilayer devices with underlayers included to enhance heater material adhesion and coating layers to limit heater damage due to bubble collapse. The heater supplies energy not only to itself and the ink but also to the adjacent coating layers. Figure 1(a) shows a typical inkjet actuator in cross section [4]. Memjet has developed two different thermal inkjet actuator technologies. The first heater design, used in Memjet Versapass printheads [5], is fully suspended in ink, allowing the bubble to form on all sides of the heater. The second design, used in Memjet Duralink and Duraflex printheads [6,7], is bonded to the floor of the chamber in a similar manner to standard devices. In the Memjet actuators, the ink layer thickness above the heater is small [8], and so instead of the bubble collapsing onto the heater, it vents to ambient [9]. Thus, coatings for cavitation protection are unnecessary, and the devices are more energetically efficient. Scanning electron micrographs of the two heater designs used in Memjet printheads are shown in Figs. 1(b) and 1(c).

Actuation of the heater is effected by a voltage pulse, which causes an electrical current to pass through the circuit, resulting in Joule heating of the electrically conductive elements. The heater is in contact with a working liquid, typically ink, and heat is transferred from the heater to the liquid, causing the liquid temperature to rise. The pulse must remain on until a bubble forms, and the temperature at which this occurs depends on the heating rate, together with the material properties of the heater and coatings, if present, and the ink. The duration of the voltage pulse, the pulse-width, $τ$⁠, which causes the Joule heating, is a key parameter that defines the operation of these devices.

When a liquid is raised above its saturation temperature, there is a finite probability that the liquid will undergo a transition to the vapor phase [10,11]. This phase transition occurs because of the formation and growth of so-called vapor embryos, which are initiated by density fluctuations in the liquid. For a liquid in a given thermodynamic state, there is an unstable equilibrium embryo radius, $re$ such that if an embryo is formed with size $r<re$⁠, it will collapse; conversely, embryos formed with $r>re$ will grow. Bubble formation is strongly affected by the thermophysical properties of the liquid of interest. It also depends upon whether nucleation occurs entirely within a region of superheated liquid, or at a boundary between two phases, typically liquid–solid. These two situations are referred to as homogeneous and heterogeneous nucleation, respectively.

The kinetics of nucleation of one phase inside another during phase change was originally described by considering the study by Gibbs [12] on the stability of phase change, see, for example, Volmer and Weber [13], Farkas [14], Volmer [15], Döring [16], and Becker and Döring [17]. Homogeneous nucleation occurs when the temperature in the liquid exceeds the attainable limit of superheat [10,11]. For thermal inkjet actuators, heat is transferred from the heater surface to the liquid, and so a bubble forms in the liquid surrounding the heater.

exceeds some limiting value, typically of order 10⁶$m−3$$s−1$⁠. In Eq. (1), $Nℓ=Naρℓ/M$⁠, $Na$ = Avogadro's number, $ρ$ = density, $σ$ = surface tension, M = molecular weight, $kB$ = Boltzmann constant, T = temperature, P = pressure, $η=exp {P∞−Ps(Tℓ)ρℓRℓTℓ}$⁠, and R = gas constant; the subscripts “$ℓ$” = liquid, “s” = saturation vapor, and “$∞$”= ambient.

Heterogeneous nucleation typically occurs if there are defects or roughness on the heater surface, with the bubbles arising at lower temperatures. The nucleation rates, J, for homogeneous and heterogeneous nucleation are compared in Fig. 2 for a range of different contact angles. Note that for the materials typically used in thermal inkjet actuators, the contact angle of water is less than 90 deg; for the materials in the current study, the contact angle of the ink is 10 deg or less.

It should be noted that heterogeneous nucleation is not a preferred mode of operation: saturation vapor pressure is a very strong function of temperature, so even small excursions from the bubble nucleation temperature can have a dramatic effect on the bubble pressure, giving rise to inconsistent droplet ejection behavior [21]. Liquid polarity can affect the surface wettability, and thus contact angle, which can strongly alter the vapor embryo nucleation rate, see Fig. 2. Roughness can provide nucleation sites that trigger nucleation before reaching the superheat limit [10], and so it is important to create very smooth thin films with roughness scales of order nanometers.

where $Tr=T/Tc$ is the reduced temperature, $Tc$ = critical temperature, and the subscripts “M” = maximum or threshold value, “s” = saturation value, and $ΔTrM=TrM−Trs$⁠. The values calculated using this expression showed small differences of less than 6% compared to the data. Subsequently, Lienhard and Karimi [28], and Lienhard [29] used the corresponding states principle [30] to estimate the nucleation properties of a variety of pure liquids. Avedisian and Sullivan [31] extended the corresponding states principle to predictions of liquid mixture nucleation temperature, finding differences of less than 1%.

Kwak and Panton [32] used a molecular interaction model to derive a slightly different form for the homogeneous nucleation rate, Eq. (1). They found good agreement between the model and data obtained with a range of simple liquids. This was extended by Kwak and Lee [33] to liquid hydrocarbons. In both studies, good agreement was found with earlier experimental data. Delale et al. [34] modified the nucleation theory to give better agreement with a selection of experimental data by adding a curve fitting parameter. The theory was then applied to liquids not used in the curve fitting process, with generally good agreement observed with experimental values.

The correlation of nucleation temperature with heating rate has mainly focused on water thus far, and here, we wish to compare data from a range of different liquids. To compare these data, the measured nucleation temperature will be normalized by the liquid critical temperature.

Several researchers have considered bubble nucleation on thin films in the context of inkjet printing. Allen et al. [38] presented an overview of the Hewlett-Packard (HP) development of thermal inkjet technology, discussing general features of the thermodynamics of their devices, but without discussing the experimental procedure in any detail.

Pöppel [39] measured the bubble temperature using an infrared detector that had been calibrated against the thin film heater resistance variation during dry pulsing of the heater. The thin film was obtained from a commercial printhead. The measurements suggested a range of nucleation temperatures up to 410 $°$C, which, even considering the uncertainty of approximately 30 $°$C, is above the critical point of water. In subsequent work, Runge [40] used pseudo-cinematographic visualization to estimate the nucleation time. This type of unsteady flow visualization technique relies on the cyclic nature of the flow phenomena under consideration, with the exposure instant gradually increased so that a time-sequence of images is obtained over the cycle. The light was focused onto the region of interest, and the reflected light was captured, with density variations showing up as variations in the image intensity. This visualization method is commonly referred to as shadowgraphy [41,42]. These movies were correlated with a one-dimensional conduction model to show that peak temperature scaled with temperature gradient at the liquid/solid boundary. Interestingly, Runge also observed nucleation temperatures above the critical point of water. This was ascribed to the fact that there are large temperature variations in the thermophysical properties of water that were not included in the model, and that there can be serious errors introduced because of temporal inaccuracies in the visualization frame rate.

Asai et al. [43] and Asai [44] used essentially the same experimental apparatus to visualize rapid boiling of water and methanol, respectively, on thin films coated with several different protective layers. The temperature at the liquid/solid boundary was calculated using a three-dimensional finite difference conduction code, together with a thermodynamic analysis based on the homogeneous nucleation rate, Eq. (1). For the water experiments, rapid heating caused nucleation near the spinodal temperature, whereas using a lower power pulse, it was possible to delay nucleation until almost 1 ms after the pulse was applied, with the interface temperature only reaching 160 °C. Asai et al. argued that the lower temperature was consistent with slow heating of a roughened surface for which there exists an abundance of localized nucleation sites. For the methanol experiments, the nucleation temperature was consistently near the spinodal, regardless of the pulse power. Experimental visualization data from the methanol experiments demonstrated that as the input heat transfer was increased, the formation of the main bubble becomes increasingly reliant on small bubbles forming early in the heating pulse. Note that Asai found the nucleation time to decrease with increasing heat transfer, which was consistent with the proposed theoretical model; Chen et al. [45] noted a similar result. Later work by Asai and coworkers [46] focused on measuring the bubble pressure as a function of time by monitoring acoustic emissions, and converting them to force using a predetermined acoustic transfer function. By comparing the transient force with stroboscopic visualizations, it was possible to identify peaks due to bubble formation and collapse.

Lin and Pisano [47,48] and Lin et al. [49] used low power to generate long-lived bubbles on small heaters (width less than 10 $μ$m), using a series of inert dielectric fluids and water. They also presented a simple model for the heater temperature using a conduction shape factor approach (see, for example, Lewis [50]). The results indicated that bubble nucleation was occurring at approximately 90% of the critical temperature of the working fluids. Similar experimental and theoretical results have been found for a number of different inert dielectric liquids by Oh et al. [51] and Lee et al. [52].

Bühler et al. [53] used a stroboscopic visualization system similar to that used in the earlier studies to examine the effect of surface coating on the bubble characteristics. The coating materials considered were: silicon nitride, silicon dioxide, silicon carbide, tantalum, tantalum oxide, gold, and platinum. The last two were found to have the best characteristics in terms of bubble uniformity and repeatability, but were found to have poor adhesion to the substrate. Among the other materials, higher uniformity and stability were achieved for higher power pulses. This is consistent with the earlier description of bubble growth as a function of power.

Andrews and O'Horo [54,55] performed bubble visualization employing a coherent AlGaInP laser to provide 50 ns pulses, with the pulse length limited by camera sensitivity. This is approximately half the indicated time resolution in the studies by Asai et al. [43,44]. The coherent source resulted in interference fringes, the effect of which was minimized by rotating the light beam through approximately one degree arc during the exposure so that the fringe pattern appeared to have a uniform intensity. The earlier time frames exhibited isolated nuclei that grew in number as time proceeds. Andrews and O'Horo noted that the size of these heterogeneously nucleated bubbles is limited due to the combined effects of strong thermal gradients, cooling due to bubble growth and penetration into lower energy fluid, and surface tension. Eventually, these nuclei enveloped almost the entire heater surface, with vapor nuclei forming at increasingly small cavity radii as the local temperature rises. Andrews and O'Horo suggested that the initial nucleation sites, which expand to a fixed size, and the later smaller sites merge to form the vapor film once the heater is covered. They also suggested that the liquid/vapor interface of the bubbles generated by the initial sites is still visible in the final stages. One further point noted was that for mixtures of liquids (in their case, de-ionized water and ethylene glycol) with different thermodynamic properties, bubble formation can occur in two distinct but closely spaced stages, depending on the difference in the spinodal temperatures for the liquids of interest. The data presented suggested that it is difficult to quantify the effect of this two-stage process.

Cornell showed that the experimental nucleation times, at a wide range of heat transfer levels, was well correlated with both the reliability model, and with the earlier empirical relation proposed by Runge [40]. Cornell noted that timing uncertainties in Runge's experiments could lead to significant uncertainties in the inferred nucleation temperatures. Over a wide range of heat transfer levels, the nucleation temperature, $Tn$⁠, for water was found to be approximately constant with $Tn=331 °C$⁠, in contrast to data presented by Runge, which showed a nearly linear relationship between heat transfer and nucleation temperature.

Rembe et al. [57] used pseudo-cinematography to capture bubble nucleation on HP DeskJet 500 printhead thin-films, where the nozzle plate was removed and replaced with a glass slide. No difference in nucleation time was found using the original nozzle plate and the glass slide. Experimental data for two different heating rates were presented, and these are consistent with the earlier study of Asai [44], in that increasingly small bubbles nucleate at higher heating rates. Real-cinematography, with a frame rate of 0.25 $μ$s, was used to capture motion of the liquid surface ejected when the nozzle plate was attached. It was shown that the variation of the location of the head of the meniscus with time could be used to determine the peak pressure in the system; for all the experiments, the peak pressure was estimated to be between 6.6 MPa and 7.0 MPa. If this is taken to be the average vapor pressure, then one can estimate the average vapor temperature to be between 282 °C and 286 °C. This is somewhat lower than the values that were estimated in an earlier study by the same authors [58], which focused mainly on the thermodynamics of phase change.

Poon and Lee [59] examined the effects of various input parameters on the bubble lifetime and time to the onset to nucleation by monitoring the reflected light intensity of a helium-neon laser beam directed at the heater element in a prototype printhead with the nozzle plate removed. The usefulness of the technique was found to depend on the surface state, which can vary with time due to wear, but was useful for making general comparisons.

Kuznetsov and Kozulin [60,61] estimated nucleation onset using the reflection of a laser beam that was directed at a thin film heater in water bath. The thin film came from a commercial HP inkjet cartridge. They estimated the temperature at nucleation onset using solutions to the heat conduction equation, and found that it tended to a value equal to the attainable limit of superheat as the heating rate increased. In a subsequent study [62], the effect of adding 0.1% silicon dioxide nanoparticles (average diameter approximately 150 nm) to the water was investigated using the same technique. It was found that the nanoparticles caused the nucleation temperature to decrease by 10–20 $°$C, with smaller differences observed for higher heating rates.

The studies discussed thus far evaluated temperature from a combination of theoretical modeling, experimental flow visualization, and reflected beam intensity measurements. Avedisian et al. [63] monitored thin-film heater resistance with time and, using carefully calibrated values of the temperature coefficient of resistivity (TCR) of the thin-film material, were able to directly estimate the nucleation temperature. The heaters were manufactured by HP, and are similar to those used in their printhead technology, except that the protective layers were removed to permit intimate contact between the heat source and liquid. The heater material was a tantalum-aluminum alloy with a very low negative TCR, which required the resolution of 0.016$Ω$ from a baseline heater resistance 15$Ω$⁠. Thus, both well-designed bridge/amplifier circuitry and accurate calibration of TCR were necessary. During this calibration, Avedisian et al. noted that the thin films required a finite number of burn-in cycles until a stable TCR value was established. The time at which nucleation occurred was determined by identifying the inflection point in the temperature versus time data, that is, where $d2T/dt2=0$⁠; for this purpose, a curve was fit to the temperature time-series data, and it was found the value for the nucleation time converged when 50 points were used to determine the fit. At low heating rates, the heater temperature oscillated after nucleation. Avedisian et al. argued that this was due to the cyclic growth and collapse of heterogeneously nucleated bubbles. From an examination of the heater surface, using scanning electron and atomic force microscopies, a very low density of defects was observed, with characteristic sizes in the range 4–100 nm. Thus, it seems that the heterogeneous nucleation was occurring at a few isolated sites. At higher heating rates, no oscillations were observed after nucleation. However, the temperatures were well below the spinodal limit. Avedisian et al. were able to correlate their experimental data with a thermodynamic model, similar to those described by Asai [44] and Cornell [56], by considering the effect of the contact angle. It is interesting to note that at the highest heat transfer rate considered, Avedisian et al. were able to overdrive their heater to very high temperatures, whereas at lower heating rates, this did not seem to occur. A subsequent study by Ching et al. [64] measured nucleation temperatures on platinum films. The values ranged from approximately 282–305 $°$C for heating rates of approximately 0.05–$0.7×109$ K/s. Experimental bubble visualization was then used to characterize the bubble dynamics [65], finding a difference in bubble shape with changes in heating rate. Cavicchi and Avedisian [66] used both heater resistance and bubble visualization to examine the effects of local pressure on the nucleation process. The bubble nucleation temperature was found to be unaffected by the local pressure, as the bubble pressure is about two orders of magnitude greater, whilst the effect of heating rate was significant, with bubble nucleation temperatures approaching the attainable limit of superheat. Thomas et al. [67] extended this work to examine the effect of adding wetting and nonwetting coatings on bubble nucleation in water: the former did not have a strong effect on nucleation temperature or rate of temperature rise, whereas the latter exhibited lower nucleation temperatures. There was also a noticeable effect of the number of pulses applied to the heater with coating, especially for the nonwetting coatings, with the effect of the coating gradually reducing with increasing numbers of pulses applied, until the heater temperature response was essentially the same as that for the uncoated heaters. This is consistent with the coatings being removed, presumably due to the repeated action of rapid heating, boiling and bubble collapse.

Iida et al. [68,69] and Okuyama et al. [70,71] combined heater resistance monitoring with bubble visualization to estimate the nucleation temperature in ethanol, toluene and water. The measured values for the organics was close to the value estimated using a slightly different form of Eq. (1), while for the water, the data were about 10% below the theory.

Zhao et al. [72] employed a Xenon flash-lamp to pseudo-cinematographically record bubble growth and collapse on the thin-film heaters in HP 51604A thermal inkjet actuators. The nozzle plate was removed to permit visualization, and the entire arrangement was placed in a large bath so that pressure measurements could be made using a Kistler 603B high frequency pressure sensor. Using a one-dimensional hydrodynamic equation together with the Rayleigh– Plesset [73,74] equation for bubble growth, Zhao et al. were able to estimate the vapor pressure at nucleation, directly from the experimental data, to be approximately 7 MPa. In a companion paper, Glod et al. [75] considered nucleation on small diameter (10 $μ$m) platinum wires, with the aim being to determine the nucleation temperature. By using platinum wires, they avoided the problems with measurement inaccuracies encountered by Avedisian et al. [63] due to the very low value of TCR of the thin-films used in that study. The temperature versus time data suggested homogeneous nucleation to play the dominant role. Similar to the earlier study, the incipient nucleation condition was determined by locating the inflection point in the temperature trace. Glod et al. noted that above a heating rate of $60×106$ K/s, the nucleation temperature reached a plateau at approximately $305 °C$⁠, with smaller values observed for lower heating rates; this is consistent with the observations in the earlier study [63].

Deng et al. [21,76] compared three-dimensional numerical simulation of the temperature field with bubble visualization and temperature estimation from heater resistance changes. They used platinum films with a range of different heater dimensions, with a relatively low heating rate of about $105$ K/s. The nucleation temperature in water was approximately $200 °C$⁠, much lower than the many of the earlier studies. This could be due to the lower heating rate and the relatively rough heater surface for some of the heater designs.

Hong et al. [77,78] compared three-dimensional computational fluid dynamics with experimental visualization and temperature data, for a polysilicon heater. The time at which the simulation initiated bubble growth was determined using the theoretical model described by Asai [44]. A pulsed diode laser with a 30 ns pulse-width was used for the pseudo-cinematographic visualizations. For the temperature measurements, the methodology of Avedisian et al. [63] appears to have been followed very closely, although the TCR for the heater material (polysilicon) was much greater than for TaAl, which improved the sensitivity of the measurements. Hong et al. found a range of nucleation temperatures from 290 to 310 $°$C for heating rates of 0.1–$0.3×109$ K/s.

Using a combination of heater temperature and bubble visualization techniques, van den Broek and Elwenspoek [79] considered bubble nucleation in ethanol using a variety of different platinum film heater designs. They found that sharp re-entrant corners give rise to locally higher current density and earlier bubble onset than when rounded corners are used. They also noted that the nucleation pulse-width varied as the inverse square of the power applied to the heater.

Xu and Zhang [80] combined heater temperature and bubble visualization experiments to investigate double pulsing of a platinum heater in methanol, finding nucleation to occur at approximately 90% of the critical point of methanol.

Hasan et al. [81] developed a one-dimensional model for bubble nucleation which combined analytical solutions to the heat conduction equation with Eq. (1). The model was compared with data from Iida et al. [68], Okuyama et al. [71] and Glod et al. [75], finding good agreement with the nucleation pulse length. Monde [82] found this model to compare well with the data of Avedisian [63], Kuznetsov et al. [60] and Xu and Zhang [80]. Li et al. [83] derived an expression for embryo formation rate which included nonequilibrium effects. They used this model to predict nucleation temperatures for water at a range of different heating rates, with broad agreement with earlier data at heating rates above $106$ K/s [11,18,69,75]; at lower heating rates, the predicted nucleation temperatures are much higher than observed [37].

Mitani [84] provides a summary of the procedures to follow when developing a thermal inkjet printhead. Mitani reported a burn-in effect for the thin films which were composed of tantalum, silicon and oxygen, similar to the results for tantalum-aluminum alloys reported by Avedisian et al. [63]. The critical heating rate was reported as $70×106$ K/s, which is close to the value determined by Glod et al. [75]. It is interesting to note that for heating rates above $300×106$ K/s, it was found that no droplet ejection could be achieved. It was conjectured that as the thermal boundary layer thickness is limited by conduction, then for a sufficiently high heating rate, the energy available in the liquid for the explosion is insufficient to cause ejection.

The heater thin-films considered thus far have been rectangular in shape, for reasons which might include ease of fabrication, ease of integration (for example, addition of protective coatings) and simplicity. Lee et al. [85] report the development of a new type of thin-film structure that is annular, and is deposited on top of the nozzle plate layer. This so-called “omega” heater can be fabricated on a single wafer, thus removing the need for wafer bonding which can dramatically lower device yield. Experimental visualizations, whilst not conclusive, indicated that increasing the power supplied to the heating element resulted in larger portions of the heater being covered with the bubble. Later work by Baek et al. [86] focused on attaching an annular heat-sink layer to the heating element, with the heat sink in contact with the liquid. The heat-sink material was chemical vapour deposition (CVD) diamond, which has a very high thermal conductivity: the films made by them gave values between 190 and 500 W $m−1$$K−1$⁠, comparable to high-k metals (gold, copper, silver), but less than natural diamond. The use of CVD coatings on a square heater was demonstrated, but bubble uniformity was poor at low power levels, potentially due to the CVD diamond surface roughness.

It should be noted that, in general, rectangular heaters are very much preferred over annular heaters, as the current density is much more uniform in rectangular heaters, and thus the temperature profile across the width of the heater is much more uniform. This has significant benefits for both heater lifetime and electrical efficiency: the heater failure mechanisms depend strongly on peak temperature, and local temperature depends on current density squared, so nonrectangular heaters have hot-spots that lead to early failure. Further, the hot-spots can trigger local nucleation well before nucleation occurs on the relatively colder parts of the heater, and vapor bubbles expanding from the hot-spots will thermally isolate the liquid in contact with these colder parts of the heater, thus preventing the colder parts of the heater from contributing to bubble formation, resulting in a much weaker vapor bubble and wasted electrical input power.

The data from the literature on nucleation temperature on thin films as a function of heating rate are shown in Fig. 3. This extends the correlations from earlier works [35,37] to include data from experiments which used organic liquids: ethanol, methanol, and toluene. Also shown are data from heated wires as well as the correlation from Skripov et al. [37] (dashed line) and the rule of thumb, $Tn/Tc≈0.9$⁠. The data are seen to be mostly close to the latter, showing a slight trend of increasing with the value of $dT/dt$ consistent with the former. The data for the water experiments are somewhat below the rule of thumb, especially for lower heating rates, and this may be due to the difficulty of removing dissolved gases from water. The heaters used in this study deliver heating rates of approximately $109$ K/s, and so it is expected that the nucleation temperatures will be close to $Tn/Tc≈0.9$⁠. The lack of more recent data for this type of experiment and of data at high heating rates provides justification for the current study.

The bubble nucleation pulse-width must be accurately specified by the printer software to ensure reliable and repeatable delivery of droplets. This study presents two methods for determining the pulse-width theoretically: via numerical simulation of the temperature field and subsequent determination of the nucleation instant, which has been used in several studies in the past; and, a simple analytical model developed here. Numerical simulations are performed with one-, two-, and three-dimensional models. It will be shown that the analytical and numerical results compare well with experimental data from two different actuator designs: one is fully suspended in the ink, the other is bonded to a substrate.

The nucleation time has been determined using numerical methods, described in Sec. 2.1, and an analytical equation, derived subsequently.

Three different problems are solved, assuming one-dimensional, two-dimensional and three-dimensional behavior. The one-dimensional assumption is reasonable as the thermal length scale in the liquid and substrate, for the values of pulse-width observed, is small compared to the heater width and length [79,89]. Two-dimensional simulations allow inclusion of lateral conduction effects without modeling the heat loss through the ends of the heater, whilst three-dimensional models include this effect.

The heat equation is solved using the finite element method: the one-dimensional model employed the scikit-fem finite element library [90], whilst the two- and three-dimensional models were solved using the FreeFem++ library [91]. Meshes were generated via Gmsh [92], and we used P2 line, triangle, and tetrahedral elements for the one-, two-, and three-dimensional simulations, respectively.

We have compared the simulation models against several analytical solutions to the heat equation, from Carslaw and Jaeger [87], and obtained very good agreement. For example, for the problem of heat generation in a slab [87], Sec. 3.14, we have found the one-dimensional model to agree within less than 0.1%. The problem of a sphere with prescribed heat flux at its surface [87], Sec. 9.7, has been simulated with the three-dimensional model, with differences between the model and the analytical solution of less than 0.2%. We have also varied the mesh size, with total numbers of elements in the three-dimensional simulations ranging from approximately 0.82–8.2 × 10⁶, and found that this changed the nucleation pulse-width by less than 0.1 ns. In these simulations, we found that the maximum temperature at the heater center varied by less 0.2 K (or less than 0.1%).

For the one- and two-dimensional finite element models, a Joule heating load was applied to the solid heater elements. For the three-dimensional simulations, a voltage difference is applied between the lower contact surfaces and an electrical finite-element calculation used to compute the Joule heating load throughout the heater. This electrical calculation used the same finite element formulation and mesh as was used in the heat conduction model, with appropriate changes in material properties and boundary conditions. The solution from the electrical calculation is then used as the load for the heat conduction model. In all cases, the far-field boundary condition was adiabatic.

In Fig. 4(a) the temperature on the heater surface is shown from a three-dimensional simulation, while in Figs. 5(a) and 5(b), the temperature field is shown from two-dimensional simulations of the suspended and bonded heater designs using a quarter-symmetry and half-symmetry model, respectively. These are used in Sec. 2.2 to derive an analytical equation for determining the nucleation time.

It should be noted that initial simulations showed that the nucleation pulse-width as calculated using the reliability function was the same, to within less than 0.1 ns, as estimating it based on simple criterion of the temperature exceeding 90% of the critical temperature value. So, to reduce the computational effort, the simulation results reported herein used this simple criterion instead of the reliability method.

Consider the heater element of a thermal inkjet actuator, with length $ℓh$⁠, width $wh$⁠, and thickness $th$⁠. The heater material has electrical resistivity $ρR$⁠, and thus the heater resistance is $Rh=ρRℓh/(whth)$⁠. The ends of the heater, called the contacts, are attached to the electrical circuitry in the chip, based on complementary metal–oxide–semiconductor (CMOS) technology, with both the contacts and the field-effect transistor (FET) in the CMOS having electrical resistance, denoted $Rc$ and $RF$⁠, respectively. A schematic of the suspended heater design is shown in Fig. 6; for the bonded heater design, there is no gap under the heater section.

where $RT=Rh+Rc+RF$ is the total resistance being the sum of the heater, contact, and FET resistance values, and $τ$ is the time over which the voltage is applied, called the pulse-width.

The electrical energy drives the Joule heating of the electrical components, together with the heat rise in the surrounding liquid, and for the bonded heater design, the heat rise in the substrate beneath the heater. Most of the Joule heating occurs in the heater, with only a minor amount in the contacts and FET. Three-dimensional finite element simulations of the heater and contact surrounded by ink show a nearly uniform temperature along the length of the heater, see Fig. 4(a); a similar result is obtained with the bonded heater. The rapid reduction in temperature at the ends of the heater seems to be matched by the rise in temperature in the adjacent contact regions; that is for $y/ℓh>0.5$⁠, the variation of $1−T/Tmax$ when plotted against $1−y/ℓh$ is very similar to the shortfall at the ends of the heater, see Fig. 4(b) in which the shaded areas are the same to within less than 1%.

Cross-sectional views of finite element predictions at the center of the suspended and bonded heater designs are presented in Figs. 5(a) and 5(b), respectively. Both of them show a thermal boundary layer which has grown outwards from the heater surface; the thickness of this layer grows with time, and the plot is shown at the instant just prior to nucleation. For the bonded heater design, a thermal boundary layer is also evident in the substrate layer. Figure 5(c) shows schematics of the thermal boundary layers in both cases.

Here, c = specific heat capacity, $ℓt,ℓ$ = thermal length scale in liquid, $ℓt,sub$ = thermal length scale in substrate, and $ΔT$ = temperature rise at the surface.

The value of $ℓt$ is equal to the ratio of the integral of temperature normal to the surface to the temperature at the surface; thus, it is equivalent to having a thermal boundary layer of thickness $ℓt$ at the maximum temperature. In Sec. 2.3, we derive an analytical approximation for the thermal length scale, finding that $ℓt≈βατ$⁠, where $β$ is a coefficient of proportionality, $α$ = $k/(ρc)$ is the thermal diffusivity and k is the thermal conductivity.

Here, we seek the positive root of Eq. (18) with the temperature rise being taken from the ambient to the superheat limit, that is $ΔT=Tshl−T∞$⁠.

where the conductors are taken as infinite away from the interface, which is valid for short times after the volumetric Joule heating s is turned on.

with $α≡k/ρc$⁠, which differs according to the sign of z; that is, a different length-scale is used in the similarity transformation on either side.

The material property $ρck$ is known as the thermal product [93,94]. Reasonable distal conditions are $θ(−∞)=0$ and $θ′(∞)=0$⁠.

Figure 7 shows the temperature in the ink as a function of time, as predicted using the one-dimensional model described above. The figure only shows temperature values above 100 $°$C. Also shown are the nucleation time according to Eq. (7) and the analytical estimate from Eq. (18) with the thermal length-scale obtained from Eq. (43). It can be seen that in this case, there is a good agreement between the methods, and also the experimental data, discussed below.

For the suspended heater designs, the bubble nucleation time for a range of different heater length and width values was obtained by stroboscopic visualization of the heater during the actuation, using an experimental setup similar to earlier studies [44,77], see Fig. 8(a). The experiment is controlled by a computer, which coordinates the firing of the electrical pulse applied to the thin film, the strobing of the light source, and the camera frame acquisition. The camera is an Allied Vision F-421B with $2048×2048$ pixels, operating at a frame rate of 16 fps. The process is repeatable, and frames are acquired every 14 ns during the actuation period. Figure 9 shows the heater before heating and just after nucleation when the bubble, indicated by a dashed line, has grown to cover the heater.

For the bonded heater designs, we noticed that the bubble nucleation time as estimated from stroboscopic visualization was equivalent to that estimated from droplet weight experiments [9]. A schematic of the experimental apparatus is shown in Fig. 8(b): the total mass of several thousand droplets is weighed using an accurate balance (A&D Weighing, model GH-252), with accounting for evaporation of the ink to the ambient. By varying the pulse-width, we can determine the minimum pulse-width for actuation when the droplet weight is above zero. In practice, we extrapolate from the first two pulse-width values to zero weight to infer the nucleation pulse-width. Figure 10 shows an example of a droplet weight curve as a function of pulse width. Here, the pulse width is expressed in terms of the number of clock pulses, with the clock frequency of 144 MHz. That is, each clock is equivalent to approximately 6.9 ns.

The TCR of the materials used for the heaters in the devices described herein is approximately 0.9 ppm/K. This is much less than used in other experiments which infer the heater temperature by resistance changes; for example, the thin films used by Avedisian et al. [63], Ching et al. [64], and van den Broek and Elwenspoek [79] had TCR = 160 ppm/K, 270 ppm/K, and 2700 ppm/K, respectively. Thus, detecting the temperature at nucleation would be a significant challenge using the current materials with very small values of TCR.

The total resistance is measured via the chip circuitry, and the heater dimensions are determined by scanning electron micrography. Knowing the heater material resistivity from four-point probe measurements [96] allows the determination of the sum of the contact and FET resistance values.

The suspended heater material is titanium aluminum nitride [97], with $ρh=4700$ kg $m−3$⁠, $kh=12.1$ W $m−1$$K−1$⁠, and $ch=576$ J $kg−1$$K−1$ [98]. The bonded heater material is titanium aluminum [99] with $ρh=5450$ kg $m−3$⁠, $kh=5.4$ W $m−1$$K−1$⁠, and $ch=636$ J $kg−1$$K−1$⁠. The bubble visualization and droplet weight testing was conducted with ink, and the ink properties at 25 $°$C are estimated to be $ρℓ=1050$ kg $m−3$⁠, $cℓ=4000$ J $kg−1$$K−1$⁠, and $kℓ=0.61$ W $m−1$$K−1$⁠. The value of J, Eq. (1), is calculated assuming the properties of water [20]. The substrate is an insulator with $ρsub=2200$ kg $m−3$⁠, $csub=725$ J $kg−1$$K−1$⁠, and $ksub=1.1$ W $m−1$$K−1$⁠.

Figure 11 compares the measured nucleation pulse-width values with those predicted using the simple analytical formula and the finite element calculations. Overall, the theoretical values compare reasonably well with each other, and there is fair agreement between theory and experiment, with the range of scatter being approximately $±10$%. Variations in heater dimensions are approximately 1% within a single printhead chip and chip-to-chip variations can be up to 2–3%. Heater material property variations are expected to be of the same order as the variation in dimensions. The theoretical models assume constant liquid material property values; however, results from some preliminary simulations which included temperature-dependent liquid material properties differed by less than 3% compared to the current models. The ink material properties are not well characterized, which may account for some of the differences observed between theory and experiment. The average differences, $Δ$⁠, between the predicted and measured pulse-width values are shown in Table 1, and it can be seen that overall, there is little between the methods, although it should be noted that the analytical model provides a slightly better estimate of the experimental values.

The nucleation pulse-width for uncoated suspended and bonded heater thermal inkjet actuators has been determined using one-, two- and three-dimensional finite element models, as well as a simple analytical model. These estimates compare fairly well with experimental nucleation pulse-width data estimated from bubble visualization and droplet weight experiments. This information has been used in the design of a number of printheads [5–7], which are used in several OEM printing systems for a wide variety of applications including labels, graphics, postfranking and additive manufacturing.

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.

E =

energy (J)

J =

rate of vapor embryo generation (⁠$m−3$$s−1$⁠)

k =

thermal conductivity (W $m−1$$K−1$⁠)

$kB$ =

Boltzmann constant (J $K−1$⁠)

$ℓh$ =

heater length (m)

$ℓt$ =

thermal length scale (m)

M =

molecular weight (kg $kmol−1$⁠)

$Na$ =

Avogadro's number (⁠$mol−1$⁠)

$Nℓ$ =

$Naρℓ/M$ (⁠$m−3$$s−1$⁠)

P =

pressure (Pa)

R =

gas constant (J $kg−1$$K−1$⁠)

$Rc$ =

contact resistance (⁠$Ω$⁠)

$RF$ =

FET resistance (⁠$Ω$⁠)

$Rh$ =

heater resistance (⁠$Ω$⁠)

$RT$ =

total resistance (⁠$Ω$⁠)

$R¯$ =

reliability (-)

s =

volumetric heat generation (W $m−3$⁠)

t =

time (s)

$th$ =

heater thickness (m)

T =

temperature (K)

$TCR$ =

temperature coefficient of resistivity, (⁠$Ω$$K−1$⁠)

V =

voltage (V)

$wh$ =

heater width (m)

x =

distance across heater width direction (m)

y =

distance along heater length direction (m)

z =

distance through heater thickness direction (m)

$α$ =

thermal diffusivity (m²$s−1$⁠)

$β$ =

constant of proportionality (-)

$ΔT$ =

temperature rise = $T−T∞$ (K)

$λ$ =

nucleation rate per unit time (⁠$s−1$⁠)

$ρ$ =

density (kg $m−3$⁠)

$ρR$ =

electrical resistivity (⁠$Ω$ m)

$σ$ =

surface tension (Pa m)

$τ$ =

pulse-width (s)

$ϕ$ =

contact angle (⁠$°$⁠)

bond =

bonded heater value

c =

critical or contact value

F =

FET value

h =

heater value

$ℓ$ =

liquid value

M =

maximum or limiting value

n =

nucleation value

r =

reduced value

s =

saturation value

shl =

superheat limit value

sub =

substrate value

sus =

suspended heater value

$∞$ =

ambient value