4.1 Experimental verification and parameter analysis

Figure 3.

Comparison of numerical results and experimental observations for ethanol droplet behaviour with an increased driving force. $V_{in}$ is 0.75, 1.0, $1.25~\text{m}~\text{s}^{-1}$ for the simulation, and the pulse amplitude $D_{v}$ is 50, 70, 80 V for the experiments, respectively.

A stable droplet formation process should show fewer satellites and high reproducibility as preferred for an inkjet process with uniform and stable deposition. Reis & Derby (Reference Reis and Derby2000) once raised a notion ‘the tail behind a forming droplet is now merged with the droplet immediately behind, and thus droplets can be obtained at distances acceptable for printing’ as the definition of a stable droplet, as commonly acknowledged by others (Smith et al. Reference Smith, Shin, Stringer, Derby and Reis2006; Gardini et al. Reference Gardini, Dondi, Costa, Matteucci, Blosi, Galassi and Cinotti2008; Jungst et al. Reference Jungst, Smolan, Schacht, Scheibel and Groll2016). Therefore, in the current study, an ejection process is considered stable if no more than two satellites are formed that are subsequently merged into the primary droplet before impacting onto the substrate.

Before carrying out a formal calculation, a convergent study was carried out to obtain mesh-independent results. The grid number, larger than 3e $+$ 04 with an averaged element quality of 0.989, is doubled and halved, respectively, until the simulation results become insensitive to the grid. Figure 3 shows a comparison between the experimental observations and simulation results. One can see that for the different stages, the predicted results agree well with the experimentally observed droplet patterns. Therefore, the applied models and assumptions are reasonable enough to obtain accurate results. When the driving force (the inlet velocity $V_{in}$ in the model, or the pulse amplitude $D_{v}$ in experiments) is weak, as shown in figure 3(a), the ejected droplet has no tail. The larger driving force leads to a long tail, which may develop into satellite droplets. For single-wave propagation in the experiments, a large driving force provides high energy, hence resulting in large droplets with high velocities.

As revealed by the experiments, the inlet velocity, $V_{in}$ , in (3.1) is approximately proportional to the pulse amplitude, $D_{v}$ . In other words, both parameters work in a similar way in driving the droplet formation. Therefore, it is reasonable to assume that the $D_{v}$ used in experiments is equivalent to the $V_{in}$ in modelling for the following studies. The droplet characteristics can be represented by the parameters at the rupture moment, including the rupture time, the jet length and the moving speed. The rupture moments for the droplets are critical for investigating droplet ejection and the moving process. After the droplet takes off from the nozzle, only the constant environment affects its behaviour. Therefore, the different rupture moments for the droplets are implicitly related to the droplet states.

Figure 4.

Influence of the pulse width on the jetting process for ethanol at (a) $6~\unicode[STIX]{x03BC}\text{s}$ , (b) $10~\unicode[STIX]{x03BC}\text{s}$ and (c) $12~\unicode[STIX]{x03BC}\text{s}$ . (d) Comparison of the predicted and experimentally measured droplet speed.

Figure 4 illustrates the influence of the pulse width on droplet ejection. The shape of the ejected droplet is greatly influenced by the pulse width. From figure 4(a), the elongated droplet breaks into a satellite droplet and a primary droplet. Then, the two droplets merge into one droplet again. However, as the pulse width increases from $6$ to $12~\unicode[STIX]{x03BC}\text{s}$ , as shown in figure 4(a–c), after the initial droplet breaks up, the number of satellites formed increases, and no merging of the droplets occurs. Figure 4(d) shows the droplet speed after leaving the nozzle. One can find that the experimental measurements and simulation results agree well with each other, which further validates the simulation models. The droplet speed increases until reaching a maximum value of approximately $5.5~\text{m}~\text{s}^{-1}$ for a pulse width of $15~\unicode[STIX]{x03BC}\text{s}$ . Then, the droplet speed decreases dramatically. The data show a significant difference in the measured and calculated speeds when the pulse width is outside of the range $7.5{-}22.5~\unicode[STIX]{x03BC}\text{s}$ . In practice, the hydrophilic identity of the nozzle material (glass in the experiment) may draw liquid back into the nozzle, which may further explain the discrepancy for the velocity in experiments and simulations. As shown by the experimental results, the pulse frequency has a negligible impact on droplet behaviour in the frequency range of 500–1500 Hz.

Figure 5.

The jetting process with a nozzle diameter of (a) $60~\unicode[STIX]{x03BC}\text{m}$ , (b) $70~\unicode[STIX]{x03BC}\text{m}$ and (c) $80~\unicode[STIX]{x03BC}\text{m}$ . The characteristic parameter at the rupture moment: (d) the rupture time, (e) the jet length and (f) the droplet velocity.

Figure 5 shows the ejected droplets for varying nozzle diameter. The time interval is $40~\unicode[STIX]{x03BC}\text{s}$ . The nozzle dimension is a key factor in determining the droplet size and ejection velocity. When the diameter is $60~\unicode[STIX]{x03BC}\text{m}$ (the minimum in this study), the ejected droplet shows a long tail. This tail gets shorter as the diameter increases. The rupture time, defined as the moment when the droplet detaches from the nozzle, increases as the diameter becomes larger. The tendency of the jet length, as defined in figure 2(c), is predicted to decrease with nozzle diameter. Similar behaviour is found for the influence of the nozzle diameter on the droplet velocity, which is defined as the velocity of the droplet when it detaches from the nozzle measured at the rupture point. Through analysis, it is found that the droplet moving velocity remains almost the same from the nozzle to the substrate since the distance between the nozzle and the substrate is not long enough to accelerate the droplet under gravity. Hence, the velocity at the rupture point is used to represent the droplet speed after ejection. From the above analysis, it is clear that the stability of the droplet can be improved using large-diameter nozzles.

The influence of the ink/fluid properties, such as density, viscosity and surface tension, on the ejection process is shown in figures 6–8. The tail of the ejected droplet becomes longer, and the number of satellite droplets increases, for increasing density. The stability of the ejection process decreases as the density gets larger (see figure 6). Although in real experiments the density variation has no direct influence on the droplet ejection, it does affect the acoustic velocity and volume elastic modulus, both of which are related to the waveform and amplitude of the driving force. Figure 7 shows the influence of the ink viscosities on the ejection process. The increased viscous force, therefore, slows down the droplet detachment and movement, which is favourable for a stable droplet. It is noted that the viscosity effect causes a viscous loss at the nozzle and requires more energy from the pressure pulse to generate the droplets with a certain velocity. Figure 8 illustrates the influence of surface tension on the ejection process. Surface tension is a key factor affecting the shape of the initial droplet and the deformation of the primary droplet and satellite droplets. As the surface tension is set to a large value, the tail of the droplet becomes mostly invisible, and the primary droplet turns around. It is noteworthy that the surface tension also depends on the wetting characteristics between the fluid and the nozzle, which play a role in droplet reproducibility. In summary, inks with small density, large viscosity and large surface tension are preferred from the viewpoint of droplet stability.

Figure 6.

Influence of the ink density on the ejection process. The liquid densities are (a) $500~\text{kg}~\text{m}^{-3}$ , (b) $1000~\text{kg}~\text{m}^{-3}$ and (c) $1500~\text{kg}~\text{m}^{-3}$ .

Figure 7.

Influence of the ink viscosity on the ejection process. The viscosities are (a) 5 cP, (b) 15 cP and (c) 25 cP.

Figure 8.

Influence of surface tension on the ejection process. The surface tensions are (a) $0.02~\text{N}~\text{m}^{-1}$ , (b) $0.04~\text{N}~\text{m}^{-1}$ and (c) $0.06~\text{N}~\text{m}^{-1}$ .

As discussed above, the droplet behaviour can be characterized in terms of the droplet parameters, i.e. the rupture time, jet length and droplet velocity. The number of satellite droplets is an intuitive description for evaluation of the stability of the ejected droplet. A summary of the operating parameters, including the droplet parameter and the ink properties, is listed in table 1. In the table, the symbols of ‘ $+$ ’, ‘ $-$ ’ and ‘ $=$ ’ represent ‘increase’, ‘decrease’ and ‘almost constant’, respectively. Two successive symbols denote a continuous change in the parameters, for example, ‘ $+-$ ’ indicates an increase followed by a decrease. From the data in the table, one can conclude that an increase in pulse amplitude, pulse width and liquid density is detrimental for achieving stable droplets. However, the coupled influence of the parameters cannot be obtained from the above analysis. As one can see from the table, the droplet parameters are influenced by the driving parameter and liquid properties in a complex manner. Since counting satellites or observing the characteristics of droplet motion is repetitive and excessive for the determination of droplet stability, it is necessary to adopt an effective and convenient criterion to prejudge whether or not the operating parameters are favourable for obtaining a stable droplet.

Table 1.

Variations of the droplet parameters affected by the drive parameters and liquid properties.

4.2 Stability analysis

The most popular criterion for stability analysis is based on non-dimensional parameters that combine fluid properties and droplet features. The capillary number ( $Ca$ ) represents the relative strength between the viscous force and surface tension force, the Weber number ( $We$ ) is a balance between the inertial and capillary forces and the Ohnesorge number ( $Oh$ ) relates the viscous forces to the inertial and surface tension forces. The numbers can be written as

(4.1) $$\begin{eqnarray}\displaystyle & \displaystyle Ca=\frac{v\unicode[STIX]{x1D707}}{\unicode[STIX]{x1D70E}}, & \displaystyle\end{eqnarray}$$

(4.2) $$\begin{eqnarray}\displaystyle & \displaystyle We=\frac{\unicode[STIX]{x1D70C}v^{2}l}{\unicode[STIX]{x1D70E}}, & \displaystyle\end{eqnarray}$$

(4.3) $$\begin{eqnarray}\displaystyle & \displaystyle Oh=\frac{\unicode[STIX]{x1D707}}{\sqrt{\unicode[STIX]{x1D70C}\unicode[STIX]{x1D70E}l}}, & \displaystyle\end{eqnarray}$$

where $v$ , $\unicode[STIX]{x1D707}$ , $\unicode[STIX]{x1D70E}$ , $\unicode[STIX]{x1D70C}$ and $l$ are the velocity, dynamic viscosity, surface tension, density and characteristic length, respectively. Here, the nozzle diameter $D$ is used as the characteristic length. Figure 9 shows the dependence of the three parameters on the ink properties, including density, viscosity and surface tension. $Ca$ remains almost unchanged while $We$ increases largely for increasing density. As viscosity increases, $Ca$ increases significantly, while $We$ decreases a little for large viscosity dissipation. Both $Ca$ and $We$ clearly decrease when the surface tension increases. For stability analysis, the reciprocal of the $Oh$ number ( $Z$ ), is adopted. An approximate range of $Z$ values for a stable droplet ejection is determined to be $4<Z<8$ , as given in figure 9, according to numerical observations. Stable and unstable images are drawn near the critical dash line in the figure. It is noted that a primary droplet with two or more satellite droplets, or one clear satellite droplet that remains separated during movement, is considered to be unstable.

Figure 9.

Non-dimensional parameters as a function of the fluid properties.

The ejection behaviour of the droplet, however, is dependent not only on the fluid properties but also on the operating conditions. The parameter, $Z$ , only considers the droplet properties explicitly unaffected by the driving conditions. Thus, a new non-dimensional parameter is introduced as a criterion to evaluate the influence of the driving parameter on droplet stability. The number is written as

(4.4) $$\begin{eqnarray}Pj=v\ast \frac{\text{d}t}{D},\end{eqnarray}$$

where $v$ is the droplet velocity, $\text{d}t$ is the pulse width and $D$ is the nozzle diameter. Since the numerical solutions were demonstrated to be accurate in the simulation of the droplet ejection process, many cases were further studied using different fluid properties and operating conditions. The cases with a stable droplet are used for calculation of the $Pj$ and $Z$ numbers. All the stable or unstable cases are denoted in figure 10 with the coordinates $Pj$ and $\log Z$ ( $\log _{10}Z$ ). As the printable range is extended to $1.6\leqslant Z\leqslant 40$ , which can be realized by controlling the operating conditions (Choi et al. Reference Choi, Kim, Lee, Lee and Kim2015), $\log Z$ is used to make the data distribution along the $x$ -axis more compact. The red and green regions contain the simulated unstable and stable droplets, respectively. The boundary between the stable and unstable regions is a convex curve with a maximum value of approximately 0.44. For a given fluid with a certain $Z$ value, the driving parameter and the nozzle size ( $Pj$ ) can be adjusted to generate a stable droplet. The universality of $Pj$ is further verified by changing $D$ for a constant $\text{d}t$ and changing $\text{d}t$ for a constant $D$ , with details shown in table 2 of the Appendix. The verification of $Pj$ by experiment was carried out using three different liquids: ethanol, mixture 1 (ethanol and glycerol with a ratio of 50 : 1) and mixture 2 (ethanol and glycerol mixed together in a ratio of 200 : 1). Details for these experiments are given in table 3 of the Appendix. Stable and unstable points are shown in figure 10. The trend for $Pj$ with $\log Z$ is the same as that found from simulation, which verifies the reliability of the $Pj$ criterion. However, the overall values for $Pj$ from the experiments are approximately 0.16 larger than those from simulation, which may be due to viscosity dissipation and surface wettability of the nozzle, which is not exactly considered in the modelling as in the experiments. Accordingly, for a given nozzle and driving parameters, appropriate fluids can be chosen for experiments using the criterion in figure 10 as a practical guide.

Figure 10.

Stability criterion for droplet ejection.