2.1. Microjetting device

The experimental setup to create and study the high-speed jet is shown in figure 1(a,b). The bubbles are generated from a breakdown caused by high electric fields. The breakdown threshold of distilled water is of the order of $70~\text{MV}~\text{m}^{-1}$ (Lide Reference Lide2004). This field strength can be conveniently achieved with microelectrodes and a piezoelectric igniter (commonly found in electric lighters). The electrodes are etched on a copper plated printed circuit board (PCB, 1.5 mm overall thickness, but milled down to 0.5 mm near and on the opposite side of the electrodes); the thickness of the wires is approximately $100~{\rm\mu}\text{m}$ and their tips are separated by approximately $130~{\rm\mu}\text{m}$ . The variations are due to the etching process and in the range of $\pm 20~{\rm\mu}\text{m}$ . The hole is located precisely between the two electrodes and it is laser drilled into the thinned PCB plate. The epoxy material is drilled with a Q-switch and ns-pulsed Nd:YAG laser (New wave research, wavelength 532 nm, pulse duration 6 ns, 40 mJ pulse energy). This leads to a tapered hole (approximately 7°, see figure 1 b); the diameter on the electrode side is approximately $130~{\rm\mu}\text{m}$ while the diameter at the opposite side ranges between 13 and $20~{\rm\mu}\text{m}$ .

2.2. Imaging

The electrode side is submerged into the liquid, while the opposite site remains dry. Care has to be taken that the current does not leak through the liquid; therefore, the electrodes are insulated with nail polish except for a small area near their gap, as shown in figure 1(b). By pressing the igniter a spark is generated within the liquid and quickly a vapour bubble expands with its centre located just below the hole. The bubble and jet dynamics is imaged with a macro lens (Nikor, $f$ 60 mm and a 30 mm extension tube) connected to a high-speed camera (Photron SA1.1) and the scene is illuminated with a bright fibre-coupled light source (metal halide bulb lamp, Sumita LS-M250). The camera is triggered from an oscilloscope, LeCroy HRO 64zi, 2 GHz maximum sampling rate.

By placing the PCB on a transparent acrylic box we can capture the bubble dynamics and the fast jets exiting the hole simultaneously. The box with a width of 8 mm and a length of 10 mm serves as a reservoir for deionized water and the silicone oils with viscosities of 10 cSt (MegaChem, Singapore), 50 cSt and 100 cSt (Dow Corning, USA) tested. For recording up to 360 000 frames per second (f.p.s.) the jet and the bubble dynamics can be observed simultaneously using a large $f$ -number; while for higher framing rates the field of view becomes limited and only the jet can be captured with sufficient detail.

Figure 1.

Schematic diagram of the experimental setup (not drawn to scale): (a) imaging system; (b) perspective of the high-speed camera, PCB and electrodes mounted on the acrylic.

2.3. Current measurements

The piezoelectric device induces a large current between the electrodes within a few nanoseconds, thus we measured the rise time of the discharge between the electrodes as reported by Tellini & Giannetti (Reference Tellini and Giannetti1998). Figure 2 displays a typical piezoelectric signal recorded by the oscilloscope; the sampling rate is 2.0 GHz; the time measured from the time the rising edge triggers the acquisition process until the signal has decayed and starts to recover, is approximately 29 ns. In these first 29 ns, a potential difference of 2.3 kV was measured; since we used a $50~{\rm\Omega}$ resistor, the current generated, 46 A, is high enough to locally boil the liquid and produce the cavitation bubble. The subsequent peaks are likely to be the result of resonances in the electric system which vanish ${\sim}300~\text{ns}$ after the initial discharge.

Figure 2.

A typical piezoelectric igniter signal that triggers the cavitation event. The graph shows the sub-microsecond timescale of the electric breakdown of 46 A, peak to peak in ${\sim}29~\text{ns}$ . The shaded region is shown in more detail in the inset of the figure.

3.1. Overview

Figure 3.

Bubble dynamics and the jets produced due to: (a) bubble expansion, (b) bubble collapse. The length of the bar in the upper right corner of the image is $500~{\rm\mu}\text{m}$ , time is in ${\rm\mu}\text{s}$ .

Once the igniter is pressed the camera is triggered and records the rapid events, all occurring within approximately $200~{\rm\mu}\text{s}$ . Figure 3 presents such a typical experiment recorded at 90 000 f.p.s. from which selected frames are shown. At time $t=0$ the current is applied, then the bubble expands in the liquid, i.e. below the dark stripe in the frames of figure 3. The bubble reaches its maximum size, its maximum vertical radius $R_{y}=1.0~\text{mm}$ , at $t=100~{\rm\mu}\text{s}$ , then it collapses at around $190~{\rm\mu}\text{s}$ . No after-bounces are observed indicating a mostly vaporous bubble. The shape of the bubble is hemispherical for most of the time and gets mildly deformed just prior to attaining its minimum size. Both the bubble expansion and the collapse lead to fast jets into the air above the dark stripe; the jets are indicated with arrows. The first and slower jet ( $t=11~{\rm\mu}\text{s}$ ) is driven by the expanding bubble to an average speed of $130~\text{m}~\text{s}^{-1}$ . The jet breaks up into droplets and has vanished when the second and faster jet exits the hole at time $t=200~{\rm\mu}\text{s}$ . Its averaged velocity is $270\pm 5~\text{m}~\text{s}^{-1}$ which is close to the speed of sound in air! The tip position for the velocity estimate is determined from a background subtracted frame; the measurement error is the result of a few pixels uncertainty in the tip position. The second jet is clearly driven by the collapse and shows a markedly blurred tip indicating fragmentation into a fine spray not resolved here. The pixel resolution of $14~{\rm\mu}\text{m}$ sets an upper limit of $28~{\rm\mu}\text{m}$ for the diameter of both jets at their base.

3.2. Jetting details

Figure 4.

The dynamics of (a) the primary jet, and (b) secondary jet, recorded at 360 000 f.p.s. The length of the bar in the upper left corner of the image is $500~{\rm\mu}\text{m}$ , time is in ${\rm\mu}\text{s}$ .

We now look into more detail of the jet dynamics with a high temporal resolution. Figure 4 is recorded at a higher frame rate of 360 000 f.p.s. and shows the first jet in figure 4(a) and the second jet in figure 4(b). Note the time in microseconds on the bottom of the frames. Only the central part of the bubble below the plate is visible. In figure 4(a), the tip of the first jet exhibits a constant velocity of $80~\text{m}~\text{s}^{-1}$ until $t=11.1~{\rm\mu}\text{s}$ . A breakup is observed at $t=13.9~{\rm\mu}\text{s}$ , that is when the bubble expansion has noticeably decelerated. This time is close to the growth time of the most amplified Rayleigh mode of a stationary $10~{\rm\mu}\text{m}$ diameter water column of approximately $11~{\rm\mu}\text{s}$ , see Eggers & Villermaux (Reference Eggers and Villermaux2008, (equation 59)). Thus the jet breakup is probably governed by capillary pinching (Lin & Reitz Reference Lin and Reitz1998). In particular the shape of the pinch-off seems unaffected by the shear layer, which agrees with the prediction for a low Weber number jet, i.e. the $We_{a}={\it\rho}_{a}r_{h}V_{j}^{2}/{\it\sigma}$ is approximately 1; $We_{a}$ is the Weber number based on air density ${\it\rho}_{a}$ , hole radius $r_{h}$ , jet exit velocity $V_{j}$ and liquid surface tension ${\it\sigma}$ . Thus capillary instability is dominating (Eggers & Villermaux Reference Eggers and Villermaux2008, equation 129). Note that the jets mentioned in these references are all steady whereas in this work they are transient. The initial velocity of the jet is sustained for a short period of time, as shown in figure 4; that allows us to estimate qualitatively the shape of the jet leaving the orifice.

The bubble wall velocity decreases from the second frame in figure 4(a) while the jet velocity remains constant. This suggests that momentum is imparted to the liquid inside the hole. Only later is the linear momentum of the jet reduced by viscous drag and gravity. As the bubble expansion decelerates less fluid is supplied into the jet and therefore the jet thins. From mass conservation, we estimate that only approximately 25 % of the liquid within the hole is ejected as a first jet. As the bubble reaches maximum expansion no more liquid exits the hole, as shown in figure 3, from the fifth frame onwards.

Figure 4(b) displays the much different and faster shape of the second jet. In the first frame at $t=175.0~{\rm\mu}\text{s}$ the bubble at the bottom of the plate is approaching its minimum volume. In the next frame the bubble remains cannot be discerned from the background, thus liquid is now flowing into the hole. The exit velocity of the jet determined from the following two frames is $376~\text{m}~\text{s}^{-1}$ , i.e. 9 % faster than the speed of sound (at $20\,^{\circ }\text{C}$ and 1 bar atmospheric pressure). The height of the jet at $t=183.3~{\rm\mu}\text{s}$ is 1.2 mm, and the upper part is a spray of tiny drops that cannot be resolved with the imaging system while the lower part retains a cylindrical shape. The lower liquid column starts to fragment at $t=188.9~{\rm\mu}\text{s}$ similar to that of the first jet. The spray on the tip cannot be explained with the Rayleigh–Plateau instability. The jet depicted in figure 4(b) at $t=183.3~{\rm\mu}\text{s}$ shows some similarity to the second wind regime described by Lin & Reitz (Reference Lin and Reitz1998), a fragmented tip followed by an intact liquid column; yet, the jet in the current work is transient while Lin & Reitz (Reference Lin and Reitz1998) discuss steady jets.

3.3. Jet velocity

The driving mechanism for the two jets is rather different. While the first jet is accelerated due to the high pressure during phase change of the liquid, the second and faster jet is caused by the converging and accelerating fluid during bubble collapse. It is expected that the resulting speeds are different. For the experiment the camera is triggered on the discharge current which is measured with a high-bandwidth current clamp. Recordings are taken at 360 000 f.p.s. The spatial and temporal resolution limits the uncertainty in the speed of the jet tip to $\pm 10~\text{m}~\text{s}^{-1}$ while the blurred view at the plate causes a larger spatial error of the vertical radius, $R_{y}$ , of $\pm 55~{\rm\mu}\text{m}$ .

We find clear dependence of the speeds of both jets as function of the vertical bubble size. Figures 5(a) and 5(b) show the increase of the speed for the first and second jet, respectively. Both have a minimum speed given by the smallest bubble we can produce with a radius of 0.6 mm; the highest speed is found for the largest bubbles with a radius of 1.05 mm. The first jet, figure 5(a), starts at $60~\text{m}~\text{s}^{-1}$ ; with increasing size we find a rather larger scatter and it reaches up to $225~\text{m}~\text{s}^{-1}$ . In contrast, the second jet, figure 5(b), increases approximately linearly from $200~\text{m}~\text{s}^{-1}$ to $300~\text{m}~\text{s}^{-1}$ and then shows a more scattered pattern with up to a supersonic speed of $400~\text{m}~\text{s}^{-1}$ . The variability of the electric breakdown is causing the observed variation in bubble sizes, and thus mechanical energies of the bubble. Here, we try to limit this variability by using a single device for all the data presented in figure 5; no clear dependence of the bubble radius on discharge current is found.

Figure 5.

The exit velocity of the jets as a function of the vertical bubble radius $R_{y}$ : (a) first jets; (b) second jets.

3.4. The shape of the first and second jets

Next, we compare the shape of the first and second jet using framing rates of 450 000 f.p.s. for similar exit speeds. Figures 6(a) and 6(b) show consecutive frames of the first and second jet, respectively, approximately 120 and $250~\text{m}~\text{s}^{-1}$ . The time between adjacent frames is $2.2~{\rm\mu}\text{s}$ . At these high speeds we find a disintegration of the jet tip which is distinct from the Rayleigh–Plateau instability. Although the second jet at $244~\text{m}~\text{s}^{-1}$ seems rather similar to the first one, it does not have a smooth cylindrical surface when it exits the hole. It already leaves the plate in the form of a mist; later, the tip is followed by a wavy cylindrical liquid mass. In contrast, the first jet at a similar velocity, figure 6(a) $250~\text{m}~\text{s}^{-1}$ , fragments into a mist at its tip with a smooth base. Hydrodynamic cavitation may lead to the fragmentation of the jet. The differences are discussed below.

The slower first and second jets at $125~\text{m}~\text{s}^{-1}$ and $118~\text{m}~\text{s}^{-1}$ respectively, show a weaker disturbance at their tip and a similar droplet breakup in the later frames.

Figure 6.

Comparison of first and second jets of water for two different exit speeds: (a) first jets; (b) second jets. The length of the bar in the upper left corner of the image is $500~{\rm\mu}\text{m}$ , the interframe time is $2.2~{\rm\mu}\text{s}$ .

Figure 7.

Second jets of water of increasing velocity. The bar in the upper left corner of the image is $500~{\rm\mu}\text{m}$ in length, the interframe time is $2~{\rm\mu}\text{s}$ .

The frames in figure 7 depict the dynamics of the second jet sorted according to their maximum velocities. The images in each frame are spaced $2~{\rm\mu}\text{s}$ in time. The velocity is indicated at the top of each frame, increasing from left to right and top to bottom. At the lowest velocities, $V_{j}\leqslant 118~\text{m}~\text{s}^{-1}$ the jet exits the hole with a rather stable shape, still the jet tip fragments within $8~{\rm\mu}\text{s}$ after exiting the hole. In the velocity range $140\leqslant V_{j}<169~\text{m}~\text{s}^{-1}$ the jet tip is slightly thicker than its base and the jet is developing from a column shape to a spray. Above $169~\text{m}~\text{s}^{-1}$ the tip leaves the hole already as spray and its base diameter increases with velocity. Very pronounced spraying is observed at $350~\text{m}~\text{s}^{-1}$ ; unfortunately, the imaging resolution does not allow a discussion of the droplet size, yet it is evident that more of the liquid is atomized at this high velocity.

A possible explanation for the transition from a laminar jet to a spray is the nucleation of cavitation within the microchannel. In order to clarify if cavitation occurs after the bubble collapse the device was completely submerged under water and a high-speed video recorded, see figure 8. Here, the main cavitation bubble collapses around $t=146.7~{\rm\mu}\text{s}$ ; at $t=153.3~{\rm\mu}\text{s}$ a vapour bubble emanates from the hole. This cylindrical bubble increases in length over the next two frames and collapses at $t=173.4~{\rm\mu}\text{s}$ . We can observe from these images that cavitation is not limited to the side with the electrodes.

Figure 8.

Bubble collapse with both sides immersed in water. A vapour cavity forms, after the collapse of the main bubble, on the opposite side which collapses around $t=173.4~{\rm\mu}\text{s}$ . The length of bar at the upper left side of the image is $500~{\rm\mu}\text{m}$ , time is in  ${\rm\mu}\text{s}$ .

5.1. Importance of Weber number

Figure 12.

Weber number versus Reynolds number: (a) first jets; (b) second jets. The open symbols are the tests performed with water and the filled symbols are the silicone oil tests. All the jets become unstable at $We\sim 4600$ .

Figure 12 plots Weber number ( $We={\it\rho}_{l}hV_{j}^{2}/{\it\sigma}$ ) versus Reynolds number ( $Re=hV_{j}/{\it\nu}$ ); $h$ is the nozzle exit diameter. The open symbols are the tests performed with water while the filled symbols represent silicone oil experiments. For the first jet, figure 12(a), as well as for the second jet, figure 12(b), transitional spray is found for the range of $We=4600\pm 800$ ; figure 12(a) shows a thin clear area that corresponds to $We=4600\pm 800$ . For this $We$ the jet is no longer a smooth liquid column; moreover, as $We$ increases the jet shows no-intact breakup length (i.e. it fragments at the nozzle exit), thus it leaves the nozzle as a spray; also, we observe an increase in the width of the jet at the exit of the nozzle. As pointed out by Lin & Reitz (Reference Lin and Reitz1998), beyond the first-wind regime different results have been reported with regard to the breakup length; these discrepancies appear to be associated with internal flow patterns induced by flow separation and cavitation. A long breakup length is exhibited by jets with detached flows while very short breakup lengths are found in cavitating nozzles; diesel injectors, where cavitation is likely to be present, show no-intact breakup length as the fluid exiting the nozzle is atomized when it leaves the injector (Chaves et al. Reference Chaves, Knapp, Kubitzek, Obermeier and Schneider1995).

Interestingly, our experimental results show that although the process of jetting for the first and second jets is rather different, in both cases the transition occurs at a similar Weber number. Cavitation inside the microchannel may contribute to the spray formation. Similar geometries and velocities for pulsed jets are found at injection valves for fuels (Payri et al. Reference Payri, Bermudez, Payri and Salvador2004). There, cavitation is prominently occurring and causing the transition from a laminar jet to a spray when the cavitation bubble reaching the nozzle exit expands. Cavitation may be present in both first and second jets; nonetheless, its presence is due to different mechanisms: for the first jet, a velocity of approximately $100~\text{m}~\text{s}^{-1}$ is reached within less than $3~{\rm\mu}\text{s}$ , thus the acceleration of the liquid is more than $3\times 10^{7}~\text{m}~\text{s}^{-2}$ . Therefore, enormous shear is generated at the channel walls and we can expect shear-induced cavitation (Joseph Reference Joseph1998). Another possibility could be the reflection of a shock wave from the liquid–gas interface within the channel, similar to the experiments reported by Ando, Liu & Ohl (Reference Ando, Liu and Ohl2006). Here too, a shock wave is generated following the electric breakdown which has been observed below the plate using strobe photography (not shown here). Upon reflection of this shock wave strong tension at the liquid–gas interface within the funnel may cause cavitation. With respect to the second jet, as portrayed in figure 8, cavitation can travel through the microchannel; as it reaches the nozzle exit it expands due to the lower ambient pressure. Figure 7 depicts that for jet exit velocities above $196~\text{m}~\text{s}^{-1}$ the width of the jet increases. The observation of the increase of the cone angle has been reported and attributed to cavitation previously by Chaves et al. (Reference Chaves, Knapp, Kubitzek, Obermeier and Schneider1995), Ganippa et al. (Reference Ganippa, Bark, Andersson and Chomiak2004) and Payri et al. (Reference Payri, Bermudez, Payri and Salvador2004). However, other mechanisms may also lead to jet instability such as nozzle internal flow, jet velocity profile and turbulence at the exit of the nozzle (Lin & Reitz Reference Lin and Reitz1998). In addition to that, the interface between a liquid jet and the surrounding gas is naturally unstable (Kelvin–Helmholtz instability), thus for large relative velocities droplets are stripped from the jet due to aerodynamic forces at the gas–liquid interface (Reitz & Bracco Reference Reitz and Bracco1982; Eggers & Villermaux Reference Eggers and Villermaux2008; van Hoeve et al. Reference van Hoeve, Gekle, Snoeijer, Versluis, Brenner and Lohse2010).

5.2. Energy considerations

Vogel et al. (Reference Vogel, Noack, Nahen, Theisen, Busch, Parlitz, Hammer, Noojin, Rockwell and Birngruber1999) report that up to 23 % of the laser energy can be converted into bubble energy for ns-laser pulses. Optimum focusing of the laser would allow generating a 1.4 mm radius bubble with a 5 mJ laser pulse. In our experiments the electric energy, $E_{e}$ , assuming a constant breakdown voltage is given by

We only integrate from $t_{0}=0$ to $t_{f}=29~\text{ns}$ , see figure 2. The subsequent small-amplitude oscillations are likely to be caused by oscillations from inductances and capacitances of the cabling and the piezoelectric device. $V_{e}$ is estimated from the dielectric strength coefficient of $70~\text{MV}~\text{m}^{-1}$ for water and $15~\text{MV}~\text{m}^{-1}$ for silicone oil using the measured distance between the electrodes of $130~{\rm\mu}\text{m}$ . The bubble maximum radius allows us to determine the potential energy stored in the liquid, $E_{b}$ , as

The kinetic energy of the jet is very crudely estimated as

assuming that the liquid in the jet moves at a constant velocity.

Figure 13.

Jet kinetic energy, $E_{j}$ , versus the cavitation bubble energy,  $E_{b}$ .

Applying these estimates for the energy, we obtain figure 13; note that this figure shows fewer data points than figure 12 because the tests conducted at the higher frame rates only captured the jet emerging from the hole, thus no bubble size information is available. For the tests conducted in water (see also figure 5), since the bubble grows as a hemispherical cap, the $E_{b}$ values presented in figure 13 provide a good estimate. However, for the more viscous liquids the $R_{y}$ value is over-estimated, thus the $E_{b}$ values presented for the silicone oil tests represent an upper bound of the bubble energy.

For water and low-viscosity silicone oil we see a good correlation of the jet energy, $E_{j}$ , with the liquid energy, $E_{b}$ . Interestingly, some of the larger bubbles create rather slow jets, as portrayed by the data points inside the dotted ellipse of figure 13; this may occur if these bubbles are created off-centre and then upon collapse the converging flow is not effectively driven through the hole. Only two jets were observed for the tests of the 100 cSt silicone oil and it can be seen that their kinetic energy is lower than that of the tests of 10 and 50 cSt oil. The efficiency scales with the square of the velocity, thus the conversion of the energy of the cavitation bubble to the kinetic energy of the jet is lower for the tests of higher viscosity. We found that the ratio of jet kinetic energy, $E_{j}$ , to bubble energy, $E_{b}$ , is 2.2 % for the tests of 10 cSt oil and 0.4 % for the tests of 50 cSt oil. For the water tests this ratio is 2.3 %. The conversion efficiency of our device is still considerably lower than the one found for strongly focused laser pulses (Vogel et al. Reference Vogel, Noack, Nahen, Theisen, Busch, Parlitz, Hammer, Noojin, Rockwell and Birngruber1999); less tight focusing strongly reduced the efficiency to only 10 %. Also, Fletcher & Palanker (Reference Fletcher and Palanker2001) report that ${\sim}$ 10 % of the electric energy is used to create the cavitation bubble, thus their device is four-fold more efficient than ours.

6.1. Liquid penetration into agarose

Here, we report on the capability of the device to produce jets that can penetrate into an elastic medium, for application to needle-free transport of liquids, i.e.  medication. The ability to fracture and penetrate elastic surfaces is of importance to deliver drugs transdermally. Agarose gel (1 % by weight in water) is a common phantom (Arora et al. Reference Arora, Hakim, Baxter, Rathnasingham, Srinivasan, Fletcher and Mitragotri2007; Nayar et al. Reference Nayar, Weiland, Nelson and Hodge2012) for human skin and is transparent, which allows visualization of the injection process.

Figure 14.

(a) The penetration of the first jet (top row) and second jet (bottom row) into 1 % agarose. The dotted white rectangle indicates the position of injection, the length of the black bar is $500~{\rm\mu}\text{m}$ . (b) Enhanced and magnified view of the injection. The frames are background subtracted; the injected volume is visible as a white blob. The top four images of (b) correspond to the rightmost top four images from left to right of (a). Also, the bottom four images of (b) correspond to the rightmost bottom four images from left to right of (a).

Figure 14 is from a typical experimental run revealing the injection dynamics. In this test a small block of agarose of approximate size $2~\text{mm}\times 2~\text{mm}\times 2~\text{mm}$ is placed on top of the hole before the jet is actuated. Figure 14(a) is a bright-field image sequence of the injection process. In the top row, second and third frame, a small liquid blob within the sample is visible. For contrast enhancement the marked rectangles from figure 14(a) are background subtracted and displayed in figure 14(b). The top row of figure 14(b) reveals the structural damage within the sample as a white region. Interestingly, the contrast ceases, which we speculate is from recovering of the structure; after $72~{\rm\mu}\text{s}$ the initial blob is not visible anymore.

The bottom row of figure 14(a) shows the injection of liquid after the bubble collapse, i.e. due to the second jet at $t>144~{\rm\mu}\text{s}$ . At $t=150~{\rm\mu}\text{s}$ the width of the liquid penetrating into the agarose is thicker than the original jet, thus the liquid stops penetrating at a depth of ${\sim}280~{\rm\mu}\text{m}$ but the fluid that is still incoming expands the lateral walls of the agarose sample. The volume injected remains visible in the successive frames. This permanent injection is probably a two-stage process: the primary jet opens up a fracture plane along which the second jet can invade the sample. The contrasted area, see bottom row of figure 14(b), reveals that more liquid is injected from the second jet while the penetration depth for both jets is similar.

6.2. Comparison with other needle-free injection techniques

Schramm & Mitragotri (Reference Schramm and Mitragotri2002) have studied the penetration of a jet into porcine and human skin; the jet can penetrate into the skin provided it can overcome the failure mechanics of skin. They also considered two types of criteria for skin failure. One criterion measures the jet impact to a critical local stress while a second examines the energy density to a critical energy density of failure. Later, Schramm-Baxter & Mitragotri (Reference Schramm-Baxter and Mitragotri2004) reported that the exit power of the jet is sufficient to describe how both jet penetration and lateral dispersion depend on jet velocity and nozzle diameter.

Following their approach we have an exit jet power, assuming constant velocity across the orifice, of

For their lowest power tested, i.e. 1 W, the distance from the skin surface to the maximum width of dispersion was $200~{\rm\mu}\text{m}$ . Interestingly, the first jets reported here have an exit power of 66 mW, but they can penetrate into the agarose sample. One explanation may be the difference in material properties: polyacrilamide used by Schramm-Baxter & Mitragotri (Reference Schramm-Baxter and Mitragotri2004) has a Young’s modulus of 380 kPa. Pailler-Mattei, Bec & Zahouani (Reference Pailler-Mattei, Bec and Zahouani2008) report a Young’s modulus of human skin in the range $11~\text{kPa}\leqslant E\leqslant 28~\text{kPa}$ , for a penetration depth of 1–5 mm; their measurements show that the Young’s modulus values measured tend to increase with penetration depth. Ahearne et al. (Reference Ahearne, Yang, El Haj, Then and Liu2005) report for a similar agar concentration of 1.25 % a value of approximately 17 kPa, thus in the range relevant for skin. Obviously, in a softer sample it is easier to overcome both the critical local stress and the critical energy density; however, Tagawa et al. (Reference Tagawa, Oudalov, El Ghalbzouri, Sun and Lohse2013) report successful penetration with an exit power of the jet of only 23 mW (for $30~{\rm\mu}\text{m}$ jet diameter at $40~\text{m}~\text{s}^{-1}$ ) into human skin and gelatin. Both the jets reported by Tagawa et al. (Reference Tagawa, Oudalov, El Ghalbzouri, Sun and Lohse2013) and from our experiments are considerably thinner than the ones from Schramm-Baxter & Mitragotri (Reference Schramm-Baxter and Mitragotri2004); thus besides the jetting power the critical stress may be relevant for penetration, too.

Table 1.

Comparison of recent experimental work done on high-speed microjets.

Table 1 summarizes the jet velocities and diameters used by different groups to compare the local static stress, jet kinetic energy, and jet power. The local static stress is ${\it\rho}_{l}V_{j}^{2}/2$ and the kinetic energy of the jet is $KE=({\rm\pi}/8){\it\rho}_{l}h^{2}V_{j}^{3}t_{j}$ , where $t_{j}$ is the interaction time of the jet with the surface. The velocity and diameter considered for the first three studies are those that correspond to the minimum values at which penetration was observed. Table 1 illustrates that the kinetic energy of the jets reported here are up to five orders of magnitude smaller while the jet power is two orders of magnitude smaller. The static stresses of our jets are comparable to those of Schramm & Mitragotri (Reference Schramm and Mitragotri2002). It is likely to be this large static load which overcomes the yield strength of the material. Similarly, the jets reported by Tagawa et al. (Reference Tagawa, Oudalov, El Ghalbzouri, Sun and Lohse2013) at $40~\text{m}~\text{s}^{-1}$ and $30~{\rm\mu}\text{m}$ have a power of ${\sim}1/3$ that of our jets and are able to penetrate into gelatin in a manner similar to that shown in figure 14.

Our method to create microjets has some similarity to several techniques presented in recent years which we will compare next. Fletcher & Palanker (Reference Fletcher and Palanker2001) and Fletcher et al. (Reference Fletcher, Palanker, Huie, Miller, Marmor and Blumenkranz2002) report microjets created by an electric discharge capable of delivering only 50–200 pl; however, their aim was to inject drugs into small blood vessels of only ${\sim}60~{\rm\mu}\text{m}$ . The laser-induced jets reported by Tagawa et al. (Reference Tagawa, Oudalov, Visser, Peters, Van der Meer, Sun, Prosperetti and Lohse2012, Reference Tagawa, Oudalov, El Ghalbzouri, Sun and Lohse2013) can inject much more liquid; they report a volume of approximately 10 nl delivered, of which ${\sim}$ 90 % is injected into gelatin (5 wt%). This is facilitated by the deep penetration depth of up to 5 mm. They report a minimum velocity for penetration of approximately $80~\text{m}~\text{s}^{-1}$ for a jet diameter of approximately $30~{\rm\mu}\text{m}$ ; similar values can be obtained with our jet, yet we expect an upper limit as the jets become unstable above $150~\text{m}~\text{s}^{-1}$ . Also, the volume delivered with our jets is similar, referring to figure 14(b), the fourth frame from the left, lower row, we estimate from the size of the white region of ${\sim}110~{\rm\mu}\text{m}$ in diameter a volume of ${\sim}$ 6 nl injected into agarose.

Although the jet creation mechanism of Tagawa et al. (Reference Tagawa, Oudalov, Visser, Peters, Van der Meer, Sun, Prosperetti and Lohse2012, Reference Tagawa, Oudalov, El Ghalbzouri, Sun and Lohse2013) is very different, the physics of injection is similar. They use a single thin jet that at high speeds causes skin failure, and then the thicker portion of the jet following the tip makes use of the gap to penetrate into. In the present study the first jet is crucial to produce the rupture of the surrogate skin (as seen in figure 14 a) so that the second jet can penetrate more effectively. It is likely that once the surface is ruptured later jets can penetrate easily; for instance triggering the device multiple times may allow delivering doses that correspond to realistic therapeutic volumes, see Arora et al. (Reference Arora, Hakim, Baxter, Rathnasingham, Srinivasan, Fletcher and Mitragotri2007).

The jet injection depth of Tagawa et al. (Reference Tagawa, Oudalov, El Ghalbzouri, Sun and Lohse2013) can be as large as 5 mm while the present device penetrates rather shallowly into the gel, approximately $450~{\rm\mu}\text{m}$ , see figure 14(b). Arora et al. (Reference Arora, Hakim, Baxter, Rathnasingham, Srinivasan, Fletcher and Mitragotri2007) have hypothesized that pain and bruising may be the consequence of deep penetration of jets that interact with tissue and blood capillaries; in order to reduce pain they limited the penetration depth of their jets to ${\sim}$ 200– $400~{\rm\mu}\text{m}$ of the skin. This is because the density of blood capillaries and nerves is lower in the upper (100– $200~{\rm\mu}\text{m}$ ) layer of the skin. Arora et al. (Reference Arora, Hakim, Baxter, Rathnasingham, Srinivasan, Fletcher and Mitragotri2007) report that jets of diameter 50– $100~{\rm\mu}\text{m}$ , $V_{j}>100~\text{m}~\text{s}^{-1}$ gave volume delivered per shot of 2–15 nl and a penetration depth of 200– $400~{\rm\mu}\text{m}$ . Notice that these parameters are comparable to the parameters of our jets.

To summarize, we compare in table 2 the parameters from the three previous works with the present study. For the jet velocity we estimate a lower bound for which penetration occurred since the speed of the jet cannot be reliably estimated from figure 14.

Table 2.

Comparison of transdermal drug delivery experiments using pulsed jets with the present study.

Of the studies mentioned above only Fletcher & Palanker (Reference Fletcher and Palanker2001) report that for higher jet velocities the jets were accompanied by hydrodynamic cavitation at the nozzle exit and those jets observed were not suitable for medication delivery.