2.1 Materials

For the experiments, we use mixtures of commercial hair gel from Europrofit and milli-Q water. The material is mainly a pH-neutralized aqueous solution of Carbopol (with triethanolamine). The material also includes a small amount of additional polymers (polyvinylpyrrolidone (PVP) and propyleneglycol) and features elastoviscoplastic properties (see Dinkgreve et al. Reference Dinkgreve, Paredes, Denn and Bonn2016). We use an MCR 502 Anton-Paar rheometer with a cone-and-plate configuration to characterize the material properties. To avoid slip (see Roberts & Barnes (Reference Roberts and Barnes2001) and also Meeker, Bonnecaze & Cloitre (Reference Meeker, Bonnecaze and Cloitre2004), Jalaal, Balmforth & Stoeber (Reference Jalaal, Balmforth and Stoeber2015)), we use sand-blasted surfaces with a roughness length scale of $4.2\pm 0.3~\unicode[STIX]{x03BC}\text{m}$ . The roughness size is close to the characteristic length of the microstructures of the solutions, being the diameter of the soft blobs of the polymers (see e.g. Kim et al. Reference Kim, Song, Lee and Park2003; Géraud et al. Reference Géraud, Jørgensen, Ybert, Delanoë-Ayari and Barentin2017). We first perform oscillatory tests for a stress range of $0.03~\text{Pa}\lesssim \unicode[STIX]{x1D70F}\lesssim 100~\text{Pa}$ . Figure 1(a) shows the values of the elastic storage modulus $G^{\prime }$ and of the loss modulus $G^{\prime \prime }$ both versus the stress $\unicode[STIX]{x1D70F}$ at an operating frequency of 1 Hz. At low stresses, the material is predominantly elastic ( $G^{\prime }\gtrsim 10G^{\prime \prime }$ ). The elastic moduli, however, decrease significantly at high stresses (when stress exceeds the yield stress), while the loss modulus becomes larger than the storage modulus.

We perform shear-rate-controlled tests to measure the viscous properties above the yield stress. The protocol was as follows: After placing the sample on the plate, we first pre-shear by ramping up the shear rate ( $\dot{\unicode[STIX]{x1D6FE}}$ ) from $0.01~\text{s}^{-1}$ to $500~\text{s}^{-1}$ . We then decreased the shear rate for the same range. The data were recorded during this decreasing shear-rate ramp. Note that a subsequent increasing shear-rate sweep test resulted in the same reported flow curve. Each data point is an average value over 10 s. The changes in the flow curves were negligible for waiting times longer than 10 s – in contrast to some other yield-stress fluids that require much longer waiting time (see e.g. Jalaal et al. Reference Jalaal, Cottrell, Balmforth and Stoeber2017; Hopkins & de Bruyn Reference Hopkins and de Bruyn2019). Figure 1(b) shows the flow curves of the used material. The flow curves approach a plateau when $\dot{\unicode[STIX]{x1D6FE}}\rightarrow 0$ , i.e. the stress approaches the yield stress $\unicode[STIX]{x1D70F}_{0}$ and the apparent viscosity blows up. We quantify the properties of the materials using the following constitutive model:

(2.1) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D70F}=\unicode[STIX]{x1D6FE}G_{0}^{\prime }\quad \text{if}\quad \unicode[STIX]{x1D70F}<\unicode[STIX]{x1D70F}_{0},\quad \text{otherwise}\quad \unicode[STIX]{x1D70F}=\unicode[STIX]{x1D70F}_{0}+K\dot{\unicode[STIX]{x1D6FE}}^{n}, & & \displaystyle\end{eqnarray}$$

where we assume that the material is a simple linear elastic solid below the yield stress ( $G_{0}^{\prime }$ being the storage modulus at a small deformation limit) and a Herschel–Bulkley fluid above it. Using such a constitutive law, we ignore the viscous effects below the yield stress as well as elastic effects above the yield stress (viscoelasticity). We will later comment on the elastic effects in appendix B.

Figure 1.

Rheology of the seven viscoplastic fluids analysed. Symbols and numbers refer to the different samples, whose material properties are given in table 1. (a) Storage modulus $G^{\prime }$ (symbols) and loss modulus $G^{\prime \prime }$ (lines) as functions of stress. (b) Flow curves: symbols are the experimental data and lines are the Herschel–Bulkley fits. (c) Viscous stress $\unicode[STIX]{x1D70F}-\unicode[STIX]{x1D70F}_{0}$ versus the shear rate (symbols) and their corresponding Herschel–Bulkley fits (lines).

In (2.1), $K$ and $n$ are the consistency and flow indices, respectively. We find these values by directly fitting the flow-curve data (solid lines in figure 1 b). To show the quality of the fits, we also plot the variation of the viscous stresses ( $\unicode[STIX]{x1D70F}-\unicode[STIX]{x1D70F}_{0}$ ) with $\dot{\unicode[STIX]{x1D6FE}}$ in figure 1(c) (see Katgert et al. Reference Katgert, Latka, Möbius and van Hecke2009; Lidon, Villa & Manneville Reference Lidon, Villa and Manneville2017). We note that our fits, although satisfactory for $\dot{\unicode[STIX]{x1D6FE}}>0.1~\text{s}^{-1}$ , include some error for small values of $\dot{\unicode[STIX]{x1D6FE}}$ . Alternatively, one can fit a linear curve to $\ln (\unicode[STIX]{x1D70F}-\unicode[STIX]{x1D70F}_{0})$ versus $\ln (\dot{\unicode[STIX]{x1D6FE}})$ with initially guessing and then searching for a yield stress that minimizes the error residuals (see appendix A in Katgert et al. (Reference Katgert, Latka, Möbius and van Hecke2009)). Such a method results in slightly larger values of the yield stress within the range of measurement errors ( ${\sim}5\,\%$ ). Table 1 lists the final coefficients for the samples used in our experiments. The rheological properties of our samples are similar to those reported for 0.1 %–1.5 % mass concentration of Ultrez 10 Carbopol in water (Dinkgreve et al. Reference Dinkgreve, Paredes, Denn and Bonn2016). For increasing polymer concentration, the elastic modulus and the yield stress increase, i.e. the gel becomes stiffer. Table 1 also lists the values of the yield stress from the stress sweep tests, where $G^{\prime }$ and $G^{\prime \prime }$ intersect (another way to measure the yield stress). The yield-stress values from the intersection of the elastic moduli curves in the stress sweep tests are larger than those found by the Herschel–Bulkley fits as also reported in Dinkgreve et al. (Reference Dinkgreve, Paredes, Denn and Bonn2016). In the text, we use the values obtained by the Herschel–Bulkley fit as that finds the most accurate values of yield stress (see the discussion in Dinkgreve et al. (Reference Dinkgreve, Paredes, Denn and Bonn2016)).

Table 1.

Values of yield stress $\unicode[STIX]{x1D70F}_{0}$ from the Herschel–Bulkley (HB) fits and from the intersection of $G^{\prime }$ and $G^{\prime \prime }$ in stress sweep (SS) tests, consistency index $K$ , flow index $n$ and storage modulus ${\mathcal{G}}_{0}^{\prime }$ , from (2.1).

Measuring the surface tension of yield-stress materials is known to be challenging. The classic methods such as capillary rise or pendant drop methods fail when the yield stress is high enough to govern the interface shape (see Boujlel & Coussot Reference Boujlel and Coussot2013; Jørgensen et al. Reference Jørgensen, Le Merrer, Delanoë-Ayari and Barentin2015). Nonetheless, the values of surface tension reported for (different types of) Carbopol in the literature (Hu, Wang & Hartnett Reference Hu, Wang and Hartnett1991; Ishiguro & Hartnett Reference Ishiguro and Hartnett1992; Manglik, Wasekar & Zhang Reference Manglik, Wasekar and Zhang2001; Boujlel & Coussot Reference Boujlel and Coussot2013; Jørgensen et al. Reference Jørgensen, Le Merrer, Delanoë-Ayari and Barentin2015) show only small deviations from that of water. We hence use the standard value for water at $20\,^{\circ }\text{C}$ , $\unicode[STIX]{x1D70E}=0.072~\text{N}~\text{m}^{-1}$ . Note that we could only measure the surface tension at very low gel concentrations (using a pendant drop method), where $\unicode[STIX]{x1D70E}=0.069\pm 0.02~\text{N}~\text{m}^{-1}$ was obtained. The slightly smaller values could be due to the presence of the other polymers inside the hair gel, which might even result in smaller surface tension values at higher gel concentrations. Therefore, the value of $0.072~\text{N}~\text{m}^{-1}$ should be taken as an upper bound for the real surface tensions of our samples. The density of the gels is just slightly ( ${\sim}0.2\,\%$ ) larger than that of water (measured using a DMA-35 Anton-Paar density meter).

2.2 Experimental set-up and parameters

Figure 2 shows a schematic of our experimental set-up. We trigger a pulsed green Nd:YAG laser (Litron Nano S65-15PIV) with a delay generator (BNC Model 575). The emitted laser beam has a diameter of 4 mm, a pulse duration of $t_{p}=6~\text{ns}$ and a wavelength of $\unicode[STIX]{x1D706}=532~\text{nm}$ . We control the energy of the laser ( $0.5~\text{mJ}<E<6.5~\text{mJ}$ ) with a $\unicode[STIX]{x1D706}/2$ wave plate, a polarizing beam splitter (PBS) and a beam dump (BD). We measure the pulse energy for every single experiment using a Gentec QE8SP-B-BL energy meter (EM). Note that, using a second energy meter (Gentec QE12LP-H-MB), we found the calibration relationship between the energy of the pulse inside the film and those measured. The laser pulse passes through a 10 $\times$ objective (Thorlabs LMH-10X-532) and is focused inside the gel layer. Using a Gaussian beam approximation we find a spot size diameter of $3.4~\unicode[STIX]{x03BC}\text{m}$ . To form the gel layers, we use spacers of given height and flatten a previously deposited blob of the material, using a blade with a thickness of $H=1~\text{mm}$ (see figure 2 b). The substrate is a glass slide clipped into a motorized 2-D stage. The yield stress of the materials prevents further spreading of the liquids. We can adjust the vertical position of the objective lens with a motorized stage, to control the focal spot $z_{f}$ . The side views are imaged with a high-speed camera (Photron FASTCAM SA-X2) attached to a long-distance microscope (Navitar 12X). The imaging is performed at a frame rate of $100\,000$ frames per second and a resolution of $17.8~\unicode[STIX]{x03BC}\text{m}$ per pixel. The illumination uses a collimated white light source (Sumita LS-M352A), passing through a diffuser. The set-up also utilizes a low-power continuous red laser for alignment and calibration of the vertical position (not shown here). The control parameters are the focal height $z_{f}$ , the laser energy $E$ and the rheological properties, e.g. the yield stress  $\unicode[STIX]{x1D70F}_{0}$ .

Figure 2.

(a) Experimental set-up for the investigation of LIFT of viscoplastic materials (dimensions are not to scale). A laser pulse passes through a $\unicode[STIX]{x1D706}/2$ plate and a polarizing beam splitter (PBS), which serve to control the beam energy. We measure the energy of the laser with an energy meter (EM). The laser is guided through a 10 $\times$ objective lens (OL) and focused on a specified focal position. The side view is imaged with a high-speed camera attached to a long-distance microscope (LDM). The test section is illuminated through backlighting. The camera is protected with a notch filter. In the schematic, BD is the beam dump. (b) Magnified view of the test section. The grey part with thickness ${\mathcal{L}}$ is the cavity at the focal height $z_{f}$ ; $D_{s}$ is the spot size; and the dashed line shows the deformed interface with a characteristic early velocity of  $U_{0}$ .

4.1 Characteristic jet velocity

One of our main goals is to categorize the jetting regimes with non-dimensional groups to find optimal conditions for printing (a clean straight jet). To do so, a characteristic velocity is required that must be known prior to the specific experiments. Here, we provide a simple analysis to find such a velocity valid for small times, i.e. early on in the jetting process. We assume the absorbed energy contributes to optical breakdown (plasma formation) and subsequent bubble growth. Based on this assumption, the balance between the energies before and after the absorption reads

(4.1) $$\begin{eqnarray}\displaystyle (I(t)-I_{p})\,\text{d}t=p(t)\,\text{d}{\mathcal{L}}, & & \displaystyle\end{eqnarray}$$

where $I$ (measured in units of $\text{J}~\text{m}^{-2}$ ) is the energy absorbed by the film per unit surface, $I_{p}$ is the plasma threshold fluence, $p$ is the pressure and ${\mathcal{L}}$ is the length of the cavity (see figure 2 b). We assume that the cavity grows at a constant speed of $\text{d}{\mathcal{L}}/\text{d}t=p/(\unicode[STIX]{x1D70C}c)$ , where $\unicode[STIX]{x1D70C}$ and $c$ are the density and the speed of sound, respectively (Fabbro et al. Reference Fabbro, Fournier, Ballard, Devaux and Virmont1990; Asay & Shahinpoor Reference Asay and Shahinpoor2012). If we also assume that the laser fluence remains constant during the pulse, we can find a characteristic pressure, namely

(4.2) $$\begin{eqnarray}\displaystyle p_{c}=(I-I_{p})^{1/2}(\unicode[STIX]{x1D70C}c)^{1/2}. & & \displaystyle\end{eqnarray}$$

This equation indeed gives the same order of magnitude ( ${\sim}10~\text{MPa}$ ) of the maximum pressures as reported in experiments with nanosecond lasers (Lauterborn & Vogel Reference Lauterborn and Vogel2013). Here we assume $c=1500~\text{m}~\text{s}^{-1}$ , as for water, since the polymer concentration is low and therefore its effect on the speed of sound is negligible (Povey Reference Povey1997; Parker & Povey Reference Parker and Povey2012).

Note that $p_{c}$ is a constant characteristic pressure that gives a simple scaling with respect to the change of the laser energy. We will later elaborate on the simplifications made here. We now assume that $p_{c}$ , over the characteristic time $t_{p}$ , provides the momentum to a material cylinder above it. Hence, $p_{c}At_{p}=Az_{f}\unicode[STIX]{x1D70C}U_{0}$ , where $A$ is the area of the spot size of diameter $D_{s}$ . We eventually find the characteristic velocity $U_{0}$ to be

(4.3) $$\begin{eqnarray}\displaystyle & \displaystyle U_{0}=\unicode[STIX]{x1D6FD}t_{p}\left(\frac{c}{\unicode[STIX]{x1D70C}}\right)^{1/2}\frac{(I-I_{p})^{1/2}}{z_{f}}. & \displaystyle\end{eqnarray}$$

In (4.3), we have included an unknown parameter $\unicode[STIX]{x1D6FD}$ that will be found later by fitting the experimental results.

We should emphasize that, by using a momentum balance like the one above, we make simplifications in five main ways. First, in reality, the pressure signal is time-dependent and strongly non-monotonic: it rapidly rises, reaches a maximum and then exponentially decays (see e.g. Lauterborn & Vogel Reference Lauterborn and Vogel2013). Also, the lifetime of this pressure signal is much larger than the laser pulse duration ( $t_{p}$ ) (Lauterborn & Vogel Reference Lauterborn and Vogel2013; Wang et al. Reference Wang, Zaytsev, Lajoinie, Eijkel, van den Berg, Versluis, Weckhuysen, Zhang, Zandvliet and Lohse2018). Therefore, a larger total momentum is expected. The rather large fitting parameter that we obtain later is mostly due to this assumption. Second, we assume that the reflection, scattering and transmission of the laser energy are negligible (as suggested by Vogel et al. (Reference Vogel, Noack, Nahen, Theisen, Busch, Parlitz, Hammer, Noojin, Rockwell and Birngruber1999)). Third, upon the optical breakdown, an (almost spherical) bubble forms that pushes the fluid and the free surface above (see appendix A). This will create a more complicated flow field inside the films, in comparison to what is assumed here (Mézel et al. Reference Mézel, Hallo, Souquet, Breil, Hébert and Guillemot2009; Brasz et al. Reference Brasz, Arnold, Stone and Lister2015; Koukouvinis et al. Reference Koukouvinis, Gavaises, Supponen and Farhat2016; Jalaal et al. Reference Jalaal, Li, Klein Schaarsberg, Qin and Lohse2019b ). Fourth, we ignore the effects of temperature change inside the liquid, as it can influence the density and consequently the speed of sound. The latter is justified, as one expects that the temperature only changes within a few micrometres near the plasma point and also reaches the ambient temperature in a few microseconds (Vogel et al. Reference Vogel, Noack, Nahen, Theisen, Busch, Parlitz, Hammer, Noojin, Rockwell and Birngruber1999). Lastly, the rheological effects, such as elasticity, plasticity and viscosity, as well as the surface tension effects are assumed to be insignificant to our problem at small time. One can examine such an assertion through the corresponding time scales. The viscous time scale $t_{vis}=(z_{f}^{2}\unicode[STIX]{x1D70C}/\unicode[STIX]{x1D707})^{1/2}\approx 10^{-2}~\text{s}$ , the gravitational time scale $t_{g}=(z_{f}/g)^{1/2}\approx 10^{-3}~\text{s}$ and the capillary time scale $t_{cap}=(\unicode[STIX]{x1D70C}z_{f}^{3}/\unicode[STIX]{x1D70E})\approx 10^{-3}~\text{s}$ are all much larger than the considered early stage time scale (which is the advection time scale, $t_{ad}=z_{f}/U_{0}\approx 10^{-5}~\text{s}$ ). Similarly, the elastic relaxation time scale $t_{el}\approx (K/G_{0}^{\prime })^{1/n}\approx 10^{-3}~\text{s}$ is also much larger than the characteristic time scale of the early stage deformation. These time scales justify our inviscid assumption to find the early stage characteristic velocity (see e.g. Brasz et al. (Reference Brasz, Arnold, Stone and Lister2015) for a similar argument in the context of jet formation).

Despite the simplifications listed above, equation (4.3) successfully predicts the scaling laws previously seen in the laser–liquid interaction tests. The predicted power-law exponent $1/2$ is indeed close to the previously reported values in the context of droplet propulsion with a laser pulse (Basko, Novikov & Grushin Reference Basko, Novikov and Grushin2015; Kurilovich et al. Reference Kurilovich, Klein, Torretti, Lassise, Hoekstra, Ubachs, Gelderblom and Versolato2016). Also, the predicted $U_{0}\sim z_{f}^{-1}$ dependence is the same as experimentally observed for jet formation in a capillary tube (Tagawa et al. Reference Tagawa, Oudalov, Visser, Peters, van der Meer, Sun, Prosperetti and Lohse2012).

The application of (4.3) is restricted to the domain that excludes the boundaries. Focusing the laser close to the glass slide, i.e. $z_{f}/H\rightarrow 1$ , results in solid ablation. This became evident when we found cracks in the glass slide. Moreover, equation (4.3) becomes singular when $z_{f}/H\rightarrow 0$ (i.e. focusing close to the free surface) and obviously can then no longer hold. In our experiments, the divergence of the velocity appears as an explosive splash of the film in the sheet opening and splash regime (see figure 4). Figure 6(a) shows the variation of the early jet velocities $U_{0}$ for different samples at different laser energies. As expected, we do not observe any significant effect of the rheological properties on $U_{0}$ , meaning that inertia is dominating any possible elastoviscoplastic effects. Moreover, the value of the plasma threshold $I_{p}$ also remains insensitive to the concentration of polymers. The predicted power law (4.3) is in good agreement with the trends observed in the experiments $U_{0}\propto (I-I_{p})^{1/2}$ . Fitting through the mean values of the data (see figure 6 a), we find $\unicode[STIX]{x1D6FD}\approx 174$ .

We then compare the model with experiments of samples at different focal heights. An example is shown in figure 6(b), where $\unicode[STIX]{x1D6FD}=174$ is fixed. The results of the model are in good agreement, implying that $U_{0}\sim z_{f}^{-1}$ is satisfactory. Note that we cannot compare the model for $z_{f}=0$ , as then $U_{0}\rightarrow \infty$ . Furthermore, we use an average value for the sound speed $c$ for the case of $z_{f}=H$ , as the plasma forms at the interface of the glass and the gel. Nonetheless, the difference between the experiments and theory remains large, which might be because of additional complexities due to the glass ablation, as explained above.

Figure 6.

Jet velocity at different laser fluences for (a) different samples, but fixed $z_{f}=0.5$ , and (b) sample 5, at different focal heights, $z_{f}$ (see legend). The yellow vertical dashed lines denote $I=I_{p}$ . In both panels (a) and (b), the solid lines are from (4.3) with $\unicode[STIX]{x1D6FD}=174$ . Error bars denote the standard deviation from multiple experiments.

4.2 Phase diagram and non-dimensional groups

We identify six different regimes of jets from our experiments, namely a bump regime (the jet is shallow with a maximum aspect ratio of less than unity), a jet regime (when a long straight jet with no significant crown forms), a crown jet regime (where a crown follows the straight jet), an unstable crown jet regime (where the edge of the crown becomes unstable and forms fingers), a fragmented jet regime (where the jet disintegrates into smaller droplets in the early stage of formation), and finally a spray regime (where the sheet opens, forms many small droplets and the sheet closure results in even more fragmentation). In figure 7, we show examples of these regimes and the dimensional phase space in these control parameters: laser energy, yield stress (gel stiffness) and focal height. When $z_{f}=0$ , we only observe sprays. As a trend, increasing the gel stiffness and/or the focal height, as well as decreasing the laser energy, results in a less developed morphology.

Figure 7.

(a) Examples of the jetting regimes observed in the experiments. The experimental conditions from top to bottom are: bump, $E=2.1~\text{mJ}$ , $z_{f}/H=0.75$ , sample no. 2; jet, $E=2.7~\text{mJ}$ , $z_{f}/H=0.5$ , sample no. 1; jet with a crown, $E=2.6~\text{mJ}$ , $z_{f}/H=0.25$ , sample no. 5; jet with an unstable crown, $E=6.3~\text{mJ}$ , $z_{f}/H=0.75$ , sample no. 6; fragmented jet, $E=6.4~\text{mJ}$ , $z_{f}/H=0.25$ , sample no. 6; spray, $E=6.3~\text{mJ}$ , $z_{f}/H=0$ , sample no. 5. (b) Three-dimensional phase space of the regimes for different laser energy $E$ , yield stress $\unicode[STIX]{x1D70F}_{0}$ and focal height $z_{f}$ . Crosses denote the experiments in which no deformation was detected. See the supplementary material for movies for the examples shown here and the 3-D animation of the phase space.

In general, inertial forces due to the laser pulse compete with capillary, viscous and plastic forces to define the dynamics and morphology of the jets. With the characteristic velocity found above (4.3), we define the following two non-dimensional groups:

(4.4a,b ) $$\begin{eqnarray}\displaystyle & \displaystyle We=\frac{\unicode[STIX]{x1D70C}U_{0}^{2}z_{f}}{\unicode[STIX]{x1D70E}}\quad \text{and}\quad Re=\frac{\unicode[STIX]{x1D70C}U_{0}^{2}}{K(U_{0}/z_{f})^{n}+\unicode[STIX]{x1D70F}_{0}}. & \displaystyle\end{eqnarray}$$

The Weber number $We$ compares the inertial stresses and capillary pressure. The Reynolds number $Re$ compares the inertial stress and the total internal stress. The latter contains two parts due to the shear thinning (with a characteristic shear rate of $U_{0}/z_{f}$ ) and the plastic viscosities. A Reynolds number of this type has previously been used in the context of viscoplastic droplets (Blackwell et al. Reference Blackwell, Deetjen, Gaudio and Ewoldt2015; Jalaal, Kemper & Lohse Reference Jalaal, Kemper and Lohse2019a ) and other configurations (Thompson & Soares Reference Thompson and Soares2016). Note that by using such a non-dimensional group, we assume that the yield stress only contributes via the plastic viscosity. We also ignore the effect of the film thickness $H$ and assume that the height of the laser focus $z_{f}$ is the only important length scale.

Figure 8 shows the non-dimensional phase diagram. Note that, in figure 8, we do not include data for $z_{f}/H=1$ , due to the complexities from the glass ablation, nor those for $z_{f}/H=0$ , as they only result in spray formation and our theoretical velocity (therefore, both $Re$ and $We$ ) diverges in that asymptotic limit. At relatively low Weber and Reynolds numbers, the inertial effects are overwhelmed by the viscous and capillary forces and only a bump forms. Increasing the Reynolds and Weber numbers results in jets, crown formation, unstable crowns and eventually fragmentation.

Figure 8.

Phase space in terms of Reynolds and Weber numbers. The dashed grey lines correspond to $Ca=0.1$ and 1. The horizontal black line shows $Re=200$ . The shaded region highlights the optimal printing condition with LIFT.

We inspect the non-dimensional map using two other non-dimensional numbers common in printing technologies: the capillary number $Ca=We/Re$ and the Ohnesorge number $Oh=We^{1/2}/Re$ . Our experimental data are bounded within the two bounds of $Ca=1$ and $Ca=0.1$ . This is a consequence of the range of experimental parameters employed here. Using (4.3), for a given gel, one finds $Ca\sim (I-I_{0})^{n/2}z_{f}^{1-2n}$ . For our gels, $2n\approx 1$ ; therefore, $Ca$ is a function of only the laser energy. For the range of laser energy we used here, $(I-I_{0})^{n/2}$ remains close to unity. Therefore, $Ca$ is almost constant for a given gel. Therefore the straight line envelopes show up. The dashed grey lines in figure 8 denote these bounds.

The Ohnesorge number (in its simple form for Newtonian liquids) is commonly used in inkjet printing application to find the regimes in which one single droplet forms (typically $0.1<Oh<1$ ). If $Oh$ is large, then the material is too viscous to be printed; and if $Oh\leqslant 1$ , then multiple droplets might form (Derby Reference Derby2010; McKinley & Renardy Reference McKinley and Renardy2011). We use the same analogy here to find the optimal regime of printing, where a straight jet forms. All the experiments shown in figure 8 feature $Oh<1$ (a common criterion in inkjet printing noted by the red dashed line in figure 8). For the range of $0.075<Oh<0.2$ , we obtain jet morphologies that can be used for LIFT printing. Within this range, the proper balance between total viscosity and inertia does not allow for fragmentation or significant crown formation. We also introduce another criterion of $Re>200$ to exclude bumps. When $Re$ is smaller than this critical value, the viscous dissipation is strong enough to resist jet formation. The shaded region in figure 8 shows the region we suggest for optimal printing with LIFT.