1. Introduction

Laser-induced plasma-mediated bubble formation (hereafter referred to as laser bubble) is an important physical phenomenon involved in many laser-based applications, such as element composition analysis in liquid [1–3], surface cleaning [4–6], cell membrane optoporation [7–10], microfluidics applications [11–14], and laser surgery [15–20]. In many medical applications, such as laser lithotripsy [15,21,22] and laser angioplasty [16,17], the liquid is usually confined by thin elastic biological pipes such as blood vessels or ureters, and in conjunction with other stiffer boundaries like the endoscope probe or kidney stones. Consequently, bubble formation may occur within a highly confined space, characterized by strong interactions between the laser bubble and its surroundings [23].

The type and strength of interactions strongly depend on the boundary conditions surrounding the laser bubble. When a laser bubble is generated in liquid water with static pressure much higher than the ambient pressure of 1 bar, the bubble dynamics will be highly suppressed with a significant reduction in size and lifetime [3,24,25]. In this scenario, the bubble suffers from a constant elevated ambient pressure.

If the bubble is produced in a small fully confined space with volume V, the temporal variation of bubble volume V_B(t) will compress the surrounding liquid, producing a time-varying ambient pressure that directly feeds back to the bubble dynamics. The magnitude of the compression pressure strongly depends on the boundary conditions. In a confined space with rigid thick walls, the extra pressure P_e can be linked with the bulk modulus of liquid K_l by P_e = K_l·V_B(t)/V [26]. The pressure is however slightly weaker in a confined space surrounded by elastic thick walls because the volumetric deformation of elastic walls partially counteracts the volume compression of the liquid. Compared with the case of rigid walls, the compression pressure is reduced by a factor of K_c/(K_c + K_l), with K_c referring to the bulk modulus of the elastic container [26,27]. In both cases, the time-varying compression pressure leads to much faster bubble dynamics compared with the situation in free liquid.

When the elastic walls in a fully confined space are thin enough, the interactions between the laser bubble and the elastic walls become more complex and are characterized by a compression pressure during bubble expansion followed by a tensile component. Orimi et al. studied the dynamics of sub-mm laser bubbles in a fully confined milliliter-sized thin tube [28]. They found that a compression pressure was built in the liquid during the bubble expansion, resulting in a reduction of both the size and period of the first oscillation. Following the compression pressure, a negative tensile pressure was detected, which prolonged the size and the period of the second oscillation. The tensile pressure is likely induced by the vibration of the elastic thin container walls [23].

Recently, we performed detailed investigations on the dynamics of laser bubbles in a small container with elastic thin walls and a partially confined free surface [23]. The cuvette was only 8-25 times larger than the bubble in its center. Our study revealed a surprisingly strong and rich interaction between the laser bubble and its surroundings. For sufficiently large bubbles, the joint action of breakdown-induced shock wave and bubble expansion excites cuvette wall vibrations, which produce alternating pressure waves that are focused onto the bubble. This results in a prolongation of the collapse phase, modulation of the second oscillation with the first oscillation, and time-delayed re-oscillations. Numerical simulations revealed that the alternating rarefaction waves from breakdown-induced wall vibrations very likely cause the prolongation of the first bubble collapse, whereas the liquid-mass movement in the cuvette corners likely results in wall vibrations causing late re-oscillations. Simulations suggested that the rarefaction wave can be described as a sinusoidal wave superimposed by an exponential decay term [23]. A sketch of the basic principle of the interactions is depicted in Fig. 1.

 

Fig. 1. Sketch of laser-induced bubble formation in a partially confined small container and the associated secondary cavitation bubbles. (a) A pulsed laser produces a high-density plasma in the focal center that causes a phase explosion leading to shock wave emission and bubble expansion. The laser also heats the impurities in the liquid, leading to nucleation. (b) Laser bubble excites the vibration of cuvette walls that generates rarefaction waves. (c) Secondary cavitation bubbles become first visible when the tensile part of the wall-induced rarefaction wave passes through the nucleation sites. (d) Secondary cavitation bubbles are visually detectable at delayed times when the tensile wave induced by liquid-mass oscillations passes through the bubble clouds.

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In that study [23], we also observed small secondary bubbles forming in the vicinity of the laser bubble but due to the limited view of the microscopic objective we were not able to see the complete dynamics of secondary bubbles. We hypothesized that these bubbles are formed when the tensile part of the rarefaction wave passes through the nucleation sites [29,30], a classical cavitation phenomenon.

Rarefaction waves are often observed to be produced by reflecting a shock wave at a flat or curved free surface between liquid and air/vapor [29–37] or at the bubble wall [38,39]. They can also arise, when a focused shock wave is launched from a finite source, as in extracorporeal lithotripsy [40–42], or by refocusing a shock wave emitted from a laser bubble through reflection at a concave boundary with finite extension [43–46]. The tensile component of these rarefaction waves is usually very short < 0.5 µs, and the generated secondary cavitation bubbles (≈ 10 µm) are much smaller than the laser bubble [29]. In contrast, wall-vibration-induced rarefaction waves produced in a small confined container with elastic walls have a much longer tensile component (tens of µs) with smaller amplitude than tensile stress waves induced by shock wave reflection or diffraction [23,28]. Such wall-vibration-induced rarefaction waves can also cause secondary cavitation bubbles (Fig. 1) [23]. Their inception and oscillations will be correlated to the dynamics of the primary laser bubble and act back on that dynamics. This interaction has not yet been studied in the past and seems worth deeper investigations.

In this study, we use a high-speed camera with an objective with large field of view to simultaneously record the dynamics of the laser bubble and secondary bubbles in a small container. We use the equivalent radius of the single laser bubble and the total area of secondary bubble clouds as figure-of-merit to track the temporal evolution of the laser bubble and secondary bubbles, respectively. Since the laser bubble serves as the driving source for the vibration of cuvette walls and liquid-mass, by varying the laser pulse energy we can tune the oscillation time T_osc of the laser bubble as well as the excitation frequency to surroundings. To precisely determine T_osc, we employed measurements from the light-scattering method and pressure measurement by a large-area pressure transducer in the container. The transducer also serves as a piston confining the liquid surface.

We show that secondary bubbles exhibit an asymmetrical distribution along the laser beam path, which are preferentially formed on the upstream side to the laser focus than on the downstream side. We also observe a periodic appearance of secondary bubbles that are synchronous with the time instants of the prolonged collapse phase and the re-oscillations of the laser bubble. Furthermore, our study reveals a remarkable distinction between the dynamics of the firstly appeared bubble clouds and the subsequently appeared bubble clouds when the driving frequency (oscillation time) of the laser bubble varies. This enables us to distinguish the wall-induced rarefaction waves that are excited directly by the laser bubble or by bubble-excited liquid-mass oscillations.

Our work demonstrates the generation of secondary cavitation by rarefaction waves having a long tensile component with relatively small amplitude. This differs from existing methods, which use short transients with large amplitudes arising from shock wave reflection or diffraction. The current study also goes beyond our own previous work that focused on the dynamics of the laser-induced bubble in a small container and its interaction with the container wall vibrations and liquid mass oscillations in the partially confined space [23]. Here we explore the interplay between the primary laser-induced bubble and the associated secondary cavitation events. This is of interest for a better understanding of the mechanisms and side effects of intraluminal laser surgery and material processing in a partially confined geometry.

2. Experimental methods

The experimental setup is depicted in Fig. 2(a). We use a frequency-doubled Nd:YAG laser (Quantel, Q-smart 450) with a wavelength of 532 nm and pulse duration of 6 ns to initiate optical breakdown and produce laser bubbles. The laser pulses are first divided by a non-polarizing 10:90 beam splitter (Thorlabs, BSN10R). The weak part is reflected onto an energy meter (Ophir, PE50-DIF-C) to measure the pulse energy. The transmitted pulse is focused into a water-filled glass cuvette through a focusing lens (L1), which consists of an achromatic doublet with a focal length of 15 mm and a meniscus lens with a focal length of 100 mm. The pulse energy is tuned from 0.2 mJ to 4 mJ to produce bubbles with R_max up to ≈ 1 mm and the peak intensity at the focal spot exceeds 3 × 10¹¹ W/cm², which is the optical breakdown threshold for ns pulses [47].

 

Fig. 2. (a) Experimental arrangement to investigate the dynamics of laser bubble and secondary cavitation bubbles in a small glass cuvette with elastic walls and partially confined free surface. A spherical bubble is generated by focusing a ns laser pulse onto the central axis of the cuvette. Bubble oscillations and pressure evolution are detected simultaneously by high-speed photography, the light scattering method of a CW probe laser beam, and the detection of far-field pressure amplitude with a piezoelectric transducer that acts as a piston confining the cuvette surface. The transverse plane of the transducer is shown as a dashed line in the top view of the cuvette. (b) Dimensions of glass cuvette and piston, and location of the laser-induced bubble.

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Figure 2(b) shows the dimensions of the glass cuvette and the piston-like transducer, which partially confines the liquid surface. The cuvette has a height of 45 mm, 12 mm × 12 mm inner size, and 1.2 mm wall thickness. The laser bubble is produced 18 mm above the bottom of the cuvette and 9 mm beneath the water surface. The cylindric transducer has a diameter of 9.25 mm that covers 47% of the surface area, while the liquid can still move freely in the cuvette corners. The cuvette was filled with deionized water with a resistivity of 18.2 MΩ·cm (Ulupure, UPD-II-10 T), which is further filtered (0.2 µm). The liquid volume up to the transducer is 3.89 mL. To avoid acoustic reflections from the cuvette bottom, a monolayer of glass spheres with 1 mm diameter is evenly distributed on the bottom. Centering of laser focus leads to symmetric outward motion of cuvette walls and to slight focusing of the rarefaction waves at the laser focus, which maximizes the pressure amplitude at this location and maintains the approximately spherical shape of the primary laser-produced bubble [23].

As a reference for the data obtained with the small glass cuvette, we performed a series of experiments simulating bubble dynamics in free liquid. For this purpose, we used a much larger cuvette with a size of 30 mm × 30 mm × 45 mm and free liquid surface. Like for the small cuvette, the laser bubble was produced on the central axis 18 mm above the cuvette bottom and 9 mm beneath the water surface. The total water height for both cuvettes is 27 mm.

Three complementary methods are employed simultaneously to monitor the temporal evolution of bubble dynamics: high-speed photography, light scattering of a probe beam, and acoustic measurements. High-speed photography is used to record the temporal evolution of the laser bubble radius R(t) and the total area of secondary bubbles A_tot(t). The light scattering and acoustic measurements are utilized to detect the first and second oscillation times of the laser bubble with high temporal resolution, as in our previous studies [23,48].

A high-speed camera (Revealer, X213) records the bubble dynamics at framing rates of 1.67 × 10⁵ frames per second (48 pixels × 440 pixels) with an inter-frame time of 6 µs. A macro lens (Sigma Macro 105 mm F2.8 EX DG OS HSM) is mounted to the camera to capture the entire events happening in the glass cuvette up to the walls. The pixel size of the CMOS camera chip is 14.6 µm × 14.6 µm. Each image pixel corresponds to 34 µm object size, as determined by imaging a calibration scale. For illumination, we use a broad-spectrum Xenon lamp (Beijing Princess, PL-X500D) to avoid speckles. The exposure time is controlled by electronic camera gating and is 100 ns for each frame.

We use an image processing algorithm to determine the pixel number within the outline of each laser bubble and to calculate the bubble cross section and its equivalent spherical radius. For secondary bubbles, we have no access to the volume information of each bubble due to a lack of stereophotography. We therefore determine the pixel number within all secondary bubbles to evaluate the total area covered by secondary bubbles.

The oscillation times T_osc1 and T_osc2 of the laser bubble cannot precisely be retrieved from the image series with 6 µs inter-framing time. For this purpose, we use acoustic and probe beam scattering measurements. A flat piezoelectric pressure transducer (Olympus, V324-N-SU) with 9.25 mm diameter and 14-32 MHz bandwidth (central frequency 25 MHz, corresponding to 40 ns time resolution) is used for far-field measurement of the acoustic transients produced by optical breakdown and bubble collapse. The time intervals between the shock waves represent the oscillation times. However, only highly spherical bubbles produce detectable shock waves after several oscillations. Focusing the laser pulses through the plane cuvette wall leads to some spherical aberrations, elongated plasmas, and deviations from spherical shape during early bubble expansion and late collapse, even when the bubble is approximately spherical at R_max [48,49]. In the small cuvette, secondary bubbles may additionally influence acoustic transient emission by the laser bubble. Therefore, a probe beam scattering technique is additionally implemented for cross-validation of T_osc1 and T_osc2.

A cw probe beam emitted by a He-Ne laser (Thorlabs, HNL020RB) is expanded and collimated by a beam expander (Thorlabs, GBE05-A) to propagated collinearly with the pump laser beam, and focused into the cuvette. The transmitted probe light is collimated by a microscope objective (Daheng optics, GCO-2131) with 10 × magnification, 0.25 numerical aperture, and focused onto an AC-coupled amplified photoreceiver (FEMTO, 25 kHz-200 MHz bandwidth). The bubble oscillations in the focal region and the shock wave emissions lead to intensity fluctuations of the transmitted beam. The signals from photodetector and transducer are recorded using a digital oscilloscope (Rohde & Schwarz, PTE1204). A delay generator (Stanford Research Systems Inc., DG645) is used for time synchronization of pump laser, high-speed camera, and oscilloscope.

The multimodal detection method enables a precise and reliable single-shot determination of the bubble dynamics and oscillation times. This is used to perform a parametric study exploring the dynamics of the laser bubble and secondary bubbles as a function of laser bubble radius R_max by tuning the pump laser energy. Altogether, 330 events are recorded with pulse energies ranging from 0.2 mJ to 4 mJ and bubble radii ranging from 200 µm to 1 mm.

3. Results

3.1 Laser bubble dynamics in free liquid

Figure 3 shows the bubble dynamics in a large cuvette simulating free liquid for E_L = 1.81 mJ. The image series in Fig. 3(a) shows that the bubble is spherical. The temporal evolution of the bubble radius R(t) is presented in Fig. 3(b). The peak oscillation amplitude decreases from R_max1 = 750 µm in the first oscillation to R_max2 = 190 µm in the second oscillation and becomes even smaller afterwards. The radius-time evolution during the first oscillation fits well with Rayleigh-Plesset simulations [23]. The images in Fig. 3(a) show no secondary bubbles.

 

Fig. 3. Dynamics of a laser bubble produced by a 1.81 mJ pulse in a large cuvette mimicking the case of free liquid. (a) High-speed photographic pictures taken at 1.67 × 10⁵ frames per second with 6.0 µs inter-frame time. Time instants are indicated on the first four frames and then on every 4^th frame. (b) Time evolution of the bubble radius derived from (a) in comparison with Rayleigh-Plesset simulations. (c) Far-field acoustic signal. (d) Light-scattering measurement. (e) and (f) show enlarged views of (c) and (d) for time intervals at breakdown and around first and second bubble collapse. Tims axis is shifted by 6 µs for acoustic signals due to the acoustic transit time over 9 mm from the laser focus to the transducer.

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Figure 3(c) shows the far-field acoustic signal, with 6 µs transit time from the emission center to the transducer at 9 mm distance. The three peaks indicate the shock waves emitted during breakdown, at first collapse, and at second collapse, with enlarged views presented in Fig. 3(e). The time intervals between the three peaks represent the first and second bubble oscillation times T_osc1 = 140.6 µs and T_osc2 = 40.0 µs. Beyond the second collapse, no peaks are visible.

Figure 3(d) shows the complete light scattering signal and Fig. 3(f) shows enlarged views. The initial positive peak belongs to the pump laser pulse. The beam is then first blocked by the breakdown shock wave and the initially small bubble but it can later pass through the larger bubble with little attenuation. The collapsing bubble first blocks the probe beam again but in the final collapse stage, light can pass because the bubble has moved upward due to buoyancy, and the bubble is now so small that the beam path is free. During rebound, the small bubble blocks the probe beam for a longer time than after breakdown [see Fig. 3(c)]. Because of the AC coupling of the photodetector, this results in an upward shift of the average signal level. At second collapse, the probe beam is transmitted again, and a positive signal peak appears.

A comparison of the collapse times shows a strong correlation between the peaks in acoustic and optical signal, with identical results for T_osc1 and T_osc2. Therefore, we evaluate T_osc2 from the optical signal in those cases, where no clear peak in the acoustic signal can be identified.

3.2 Laser bubble and secondary bubble dynamics in the small glass cuvette

Figure 4(a) shows the bubble dynamics in a small glass cuvette with partially confined free surface [Fig. 2(b)] for a pulse energy of 1.88 mJ, similar to the energy used for the case of free liquid (Fig. 3). A spherical bubble with similar size as in free liquid is produced in the center of the cuvette but now a cloud of small secondary bubbles is visible along the laser path close to the laser bubble. The secondary bubbles are preferentially located on the upstream side of the laser bubble and extending towards the wall. Furthermore, a high-pitch sound can now be heard when the laser bubble is generated. The sound emission in the audible frequency range (< 15 kHz) likely arises from cuvette wall vibrations. We will later deduce a wall vibration frequency of 8.6 kHz from the oscillation period of the T_osc2 (T_osc1) curve in Fig. 6.

 

Fig. 4. Dynamics of laser bubble and secondary bubbles in a small glass cuvette. (a) High-speed photographic pictures taken at 1.67 × 10⁵ frames per second with 6.0 µs interframe time. Time instants are indicated on the first four frames and the subsequent every 4^th frames. (b) Temporal evolution of the laser bubble radius determined from Fig. (a). (c) Temporal evolution of the total area of the secondary bubbles. The pump laser pulse incidents from the left. Pump pulse energy E_L = 1.88 mJ, the maximum radius and first oscillation time of laser bubble are R_max = 709.0 µm and T_osc1 = 170.8 µs respectively.

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Figure 4(b) shows the time evolution of the laser bubble radius determined from Fig. 4(a). It exhibits a pronounced prolongation of the collapse phase during the first oscillation period. The measured R(t) curve starts to deviate from the Rayleigh-Plesset simulations around t = 100 µs. This results in a longer T_osc1 of 170.8 µs compared with the free liquid scenario, where T_osc1 = 140.6 µs. Moreover, we see that the laser bubble experiences periodic re-oscillations after the first oscillation, reaching a first peak at around 300 µs and a second peak at around 460 µs. The peak-to-peak distance of the two re-oscillations is ≈ 160 µs. This phenomenon is absent in the free liquid case.

The observations of a prolonged collapse phase and T_osc1 and periodic re-oscillations have already been reported in our previous study with slightly smaller cuvette [23]. The analysis of these observations revealed that the expansion of laser bubbles in a partially confined small container not only induces a pressure rise in the surrounding liquid that may reduce the maximum bubble size but it also excites vibrations of elastic cuvette walls. The wall vibrations generate pressure waves traveling toward the laser bubble in the cuvette center. The pressure wave is bipolar, starting with a compression wave from the bubble expansion followed by alternating tensile and compression waves from the wall vibrations with rapidly decaying amplitude [23]. The period of the pressure oscillations reflects the vibration period of the walls, and the amplitude of the tensile component depends on the laser bubble size. It reaches a few bars for R_max > 200 µm [23]. Additionally, the oscillation of the laser bubble also drives an oscillatory movement of the liquid-mass in the four corners of the cuvette around the transducer. That movement creates a pressure variation in the liquid with slowly-decaying amplitude. This pressure variation can likely also excite wall vibrations, which lead to the formation of delayed rarefaction waves that induce re-oscillations of the residual laser bubble in the liquid [23]. The wall vibrations (faster decay and shorter period) and the liquid-mass oscillations (slower decay and longer period) are both excited by the laser bubble oscillations. They constitute a coupled oscillation system with 3 eigen-periods, T_wall, T_lm and T_osc1, which produces various interesting phenomena in the small container [23].

The rarefaction waves generated through bubble-excited wall vibrations do not only influence the dynamics of laser bubbles but also induce small secondary bubbles in the cuvette, as seen in Fig. 4(a) and (c). Due to limitations of image resolution and the coalescence of secondary bubbles, it is quite difficult to trace the evolution of each secondary bubble. Therefore, we calculated the total area of secondary bubbles and tracked its temporal evolution, A_tot(t), to assess the influence of rarefaction waves on the behavior of bubble clouds, as shown in Fig. 4(c). Here, the area A_tot represents the 2D projection of the total volume of the bubble clouds. Since all secondary bubbles are exposed to identical rarefaction waves, they are in phase, and the temporal variation of A_tot reflects the pressure variation in the rarefaction waves.

Figure 4(c) shows three peaks in the A_tot(t) curve with similar peak-to-peak distance of ≈ 160 µs, similar to the time separation of the two re-oscillations in Fig. 4(b). The first peak appears at t ≈ 100 µs and reaches a maximum at t₁ = 136 µs [marked red in Fig. 4(a)]. Afterwards the bubble cloud collapses and completely vanishes at t = 178 µs. The peak coincides with the prolonged collapse phase of the laser bubble. The bubble cloud re-expands and the area reaches a second peak at t₂ = 298 µs [marked blue in Fig. 4(a)]. A third maximum, albeit weak, is reached at t₃ ≈ 460 µs. The time instants t₂ and t₃ correspond to the moments, when the two re-oscillations of the laser bubble reach their maximum amplitude. We conclude that the generation of secondary bubbles in the glass cuvette coincides with the collapse prolongation and subsequent re-oscillations of the laser bubble. This indicates that both phenomena are governed by the same rarefaction waves. The re-oscillations and rebounds of the secondary bubble clouds are attributed to the liquid-mass oscillations in the cuvette corners. Their time period of ≈ 160 µs is slightly larger than the value of T_lm = 120 µs found in our previous study for a smaller cuvette with l = 10 mm [23].

3.3 Energy dependence of laser bubble dynamics

Figure 5 depicts the energy dependence of the maximum radius R_max and of the first oscillation time T_osc1 of the laser bubble induced in the partially confined glass cuvette in comparison with free liquid. In free liquid, R_max scales linearly with the cube root of pulse energy, E_L^1/3 for R_max > 100 µm [Fig. 5(a)]. This corresponds to a proportionality between bubble energy ∝ R_max³ and laser pulse energy. Figure 5(b) shows that in free liquid T_osc1 scales linearly with R_max, in agreement with the well-known Rayleigh formula and previous studies [23,50].

 

Fig. 5. Energy dependence of the maximum radius R_max and first oscillation time T_osc1 of the laser bubble induced in the glass cuvette compared with those in free liquid. (a) R_max as a function of the cubic root of laser pulse energy E_L. (b) T_osc1 as a function of R_max. Rayleigh formula can be expressed as R_max = 5.46×T_osc for room conditions.

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In the partially confined glass cuvette, R_max is linearly proportional to E_L^1/3 only in the lower range of pulse energies, where R_max < 600 µm [Fig. 5(a)]. For larger bubble sizes, the R_max(E_L^1/3) curve gradually deviates from and stays below linearity. Due to the partial confinement, the bubble expansion produces a transient increase of ambient pressure proportional to the bubble volume 4/3πR(t)³ that feeds back on the expansion process. This leads to a reduction of R_max, which becomes ever larger with increasing pulse energy. This feature explains also the slight deviation of R(t) between the experimental data and simulations around maximum expansion in Fig. 4(b), where the ratio of bubble volume to liquid volume is 0.02%. The bubble-induced rise of ambient pressure was first reported for bubble oscillations in a fully confined space [23,26,28,51]. Here, we show that it plays a role also in a small container with a partially confined free surface if the bubble-to-liquid volume ratio is large enough.

Figure 5(b) shows that in the small cuvette T_osc1 scales linearly with R_max only for relatively small bubbles, where R_max < 540 µm and T_osc1 < 100 µs. For larger bubbles, the oscillation time is prolonged and the T_osc1(R_max) curve lies above the prediction of the Rayleigh formula and the data for free liquid. As seen in Fig. 4(b), the prolongation occurs exclusively during the bubble collapse phase, in agreement with our previous results [23]. Tensile waves are generated, when the cuvette walls continue to swing outward due to inertia but the laser bubble already shrinks. When the rarefaction waves generated at the walls meet the collapsing bubble, they slow down its collapse phase, which leads to the prolongation of T_osc1 [23]. Synchronous with the collapse prolongation of the laser bubble, the tensile waves expand secondary bubbles from nuclei created by laser heating of impurities [Figs. 4(b) and (c)].

Figure 6 shows the dependence of the second oscillation period T_osc2 on T_osc1. One can see that T_osc2 undulates with T_osc1, exhibiting a sine-like pattern with a spacing between the adjacent maxima of 116 µs. The modulation of T_osc2 with T_osc1 is attributed to a change of the relative phase between the laser bubble oscillation and the wall vibrations. The period of the modulation reflects the eigen-period of the wall oscillations with T_wall = 116 µs (corresponding to a frequency of 8.6 kHz). It is slightly longer than the value of T_wall = 80 µs found in our previous study for a smaller cuvette with l = 10 mm [23].

 

Fig. 6. Dependence of the second oscillation time of the laser bubble T_osc2 on the first oscillation time T_osc1 in the glass cuvette. The peak-to-peak distance in the T_osc2 (T_osc1) curve reflects the wall vibration period with T_wall = 116 µs.

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3.4 Energy dependence of secondary bubble dynamics

Figure 7 presents picture series of the bubble dynamics at various pulse energies ranging from 1.38 mJ to 3.59 mJ. Bubble size ranges from 634 µm to 865 µm, and oscillation times from 130.3 µs to 217.2 µs. At large pulse energies, slight deviations from spherical shape are observed that are probably due to spherical aberrations of the pump laser beam resulting in elongated plasmas [48,49]. Secondary bubbles are created from the vicinity of the laser bubble up to the cuvette wall. Bubble clouds exhibit several oscillation cycles following the rhythm of rarefaction waves, forming several peaks in the A_tot(t) curve of Fig. 4(c). To save space, the photos in Fig. 7 show only the first maximum expansion (first column) and the second maximum expansion (second column) of the secondary bubbles. The complete dynamics is presented in the Supplement 1.

 

Fig. 7. Selected high-speed photographic pictures of secondary bubbles at various pulse energies with (a) E_L = 1376 µJ, (b) E_L = 1804 µJ, (c) E_L = 2276 µJ and (d) E_L = 3592 µJ. Laser bubble sizes are tabulated in the figure and the corresponding T_osc1 are 140.3 µs, 173.4 µs, 182.7 µs, and 217.2 µs, respectively. All photos are taken at 1.67 × 10⁵ frames per second with 6.0 µs inter-frame time. Complete photo series for each pulse energy are presented in Supplement 1 Figs. S2-S5.

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Secondary cavitation bubble formation exhibits a threshold behavior; bubbles become visible only when the pulse energy exceeds 1.0 mJ, with R_max > 580 µm and T_osc1 > 116 µs (Supplement 1 Fig. S1). The minimum bubble radius that can be visualized in our imaging system is 17 µm. Secondary bubbles may already be created with smaller pulse energies but due to the limitation of our imaging system, they would not be visible. The threshold for the appearance of secondary bubble is consistent with the threshold for laser bubble prolongation at R_max= 540 µm [Fig. 5(b)].

Let us now look at the energy dependence of the first maximum expansion of bubble clouds. For pulse energies slightly higher than 1.0 mJ, secondary bubbles appear sparsely along the optical axis with small sizes, Fig. 7(a) left. When pulse energy increases, the number density of secondary bubbles increases and the bubbles gradually fill the entire cone angle of the laser beam, Fig. 7(b)-(d) left. Secondary bubbles are mainly observed on the upstream side of the laser bubble; only a few bubbles are formed on the downstream side.

We were not able to quantify the size distribution of each secondary bubble due to the limited resolution of our imaging system, a lack of stereophotography, and the coalescence of bubbles in the clouds. However, we can roughly evaluate that the maximum radius of secondary bubbles ranges from tens of micrometers up to ≈ 100 µm, depending on the pulse energy. We can also qualitatively see that secondary bubbles near the wall have a larger size than those close to the laser bubble, [Fig. 7(d) left]. We also observe that no secondary bubbles are visible in the intermediate vicinity of the laser bubble. Especially between the upstream part of the secondary bubble clouds and the laser bubble, a significant gap is visible. The gap and the inhomogeneous size distribution in the clouds indicate a mutual interaction between laser bubble and secondary bubble clouds.

After the first expansion, the secondary bubbles collapse rapidly but they expand again and reach a second maximum after ≈ 165 µs [Figs. 7(a)-(c)] and 220 µs [Fig. 7(d)]. Compared with the first maximum expansion stage, the bubbles are generally smaller during their second expansion. An exception is Fig. 7(a), where the opposite occurs. Another distinct difference lies in the bubble distribution: Secondary bubbles appear also in the vicinity of the laser bubble, and the bubble density on the downstream side of the laser bubble is larger during the second expansion phase than during the first expansion. This phenomenon is most obvious at large pulse energy [Fig. 7(d)]. At large pulse energies, the laser bubble migrates towards the direction of the incoming laser beam, where most secondary bubbles are produced. This is also an indication of a mutual interaction between the laser bubble and the bubble clouds.

Although we were not able to quantify the size evolution of each secondary bubble, we could evaluate the temporal evolution of the total area of the secondary bubbles, A_tot(t). Figure 8 shows the A_tot(t) curve for bubbles produced at various pulse energies ranging from 1.4 mJ to 3.6 mJ, corresponding to laser bubble size from 630 µm to 870 µm. Figure 8(a) shows cases with lower pulse energies and laser bubbles with R_max < 760 µm and T_osc1 < 200 µs, whereas Fig. 8(b) shows cases with larger bubbles with T_osc1 > 200 µs.

 

Fig. 8. Temporal evolution of the total area of the secondary bubbles A_tot(t) produced at various pulse energies. (a) Pulse energy E_L < 3.0 mJ and laser bubble oscillation time T_osc1 < 200 µs; (b) Pulse energy E_L > 3.0 mJ and bubble oscillation time T_osc1 > 200 µs.

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The A_tot(t) curves in Fig. 8(a) have three peaks corresponding to the three expansion phases of secondary bubbles. With increasing pulse energy, the maximum value of the first peak, A_tot,max1, increases monotonically from 0.83 × 10⁵ µm² at E_L = 1.38 mJ to 13.4 × 10⁵ µm² at E_L = 2.28 mJ. However, the maximum value of the second peak, A_tot,max2, exhibits an non-monotonic behavior: it first increases from 3.1 × 10⁵ µm² at E_L = 1.38 mJ to 5.4 × 10⁵ µm² at 1.68 mJ and then drops to 1.5 × 10⁵ µm at 2.23 mJ. These trends become more obvious in Fig. 9, where more measurements are presented. Along with the monotonic increase of A_tot,max1 with pulse energy, the time instant at which the secondary bubbles are firstly maximally expanded, t₁, exhibits a gradual shift from t₁ = 118 µs at E_L = 1.38 mJ to t₁ = 130 µs at E_L = 1.68 mJ until it asymptotically reaches a fixed value of 136 µs at high pulse energies. It is well known for acoustically driven bubbles that with increasing amplitude of the driving pressure they reach their maximum size at ever later times until finally an asymptotic behavior is reached, where the maximum expansion occurs at the moment when the tensile component of the driving pressure wave ends (see Fig. 4 in Ref. [33]). For the second expansion phase, the time at which the bubble clouds are maximally expanded shifts asymptotically to t₂ ≈ 298 µs. The third bubble expansion is weaker, with A_tot,max3≤ 1.8 × 10⁵ µm². Here, maximum expansion occurs approximately t₃ ≈ 454 µs. If we only consider the asymptotically reached time instants for maximum expansion, we find that the intervals between t₁ and t₂, and between t₂ and t₃ are approximately equal with 160 µs. This is consistent with the value for the period of liquid-mass oscillation, T_lm = 160 µs, that was deduced from Fig. 4(b). However, this similarity does not apply for larger laser-bubble sizes with R_max > 820 µm and T_osc1 > 200 µs. Figure 8(b) shows that t₁ remains at 136 µs but t₂ now jumps to a larger value of about 350 µs. The time interval between t₁ and t₂ extends to 214 µs, much longer than the value in Fig. 8(a).

 

Fig. 9. Maximum area of secondary bubbles at the first and second expansions, A_tot,max1,2, produced by laser pulses at various energies, which are represented by the oscillation time of laser bubbles T_osc1. A monotonic increase is observed for the A_tot,max1(T_osc1) curve which can be fitted by a growth function, whereas an undulated pattern can be seen in the A_tot,max2(T_osc1) curve, which can be fitted by a sinusoidal function. The Green dashed box indicates the region in which the time interval between t₁ and t₂ abruptly grows from ≈ 160 µs to 214 µs for laser bubbles with T_osc1 > 200 µs.

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To quantify the energy dependence of A_tot,max1 and A_tot,max2, we evaluated 60 breakdown events induced by pulse energy ranging from E_L = 1.0 mJ to 3.6 mJ, where bubble clouds are visible in our imaging system. The induced laser bubble size ranges from R_max = 580 µm to 860 µm with T_osc1 ranging from 115 µs to 220 µs. We plotted A_tot,max1,2 as a function of T_osc1 because T_osc1 represents the driving frequency of the laser bubble, as shown in Fig. 9. While the undulations of T_osc2(T_osc1) in Fig. 6 and A_tot(t) in Fig. 8 reflect the eigen oscillation period of cuvette wall and liquid-mass respectively, the peaks A_tot,max1,2 reflect their maximum oscillation amplitude. We observed two remarkable differences between the curve A_tot,max1(T_osc1) and A_tot,max2(T_osc1): the former exhibits a monotonic increase that can be fitted by a growth function, whereas the latter shows a undulating pattern that can be fitted by a sinusoidal function. The two curves cross each other at T_osc1 = 165 µs. For smaller bubbles with T_osc1 < 165 µs and R_max < 680 µm, the second maximum A_tot,max2 is even larger than A_tot,max1. The green dashed box in Fig. 9 marks the A_tot,max2 regime for large bubble with T_osc1 > 200 µs, where the time interval between t₁ and t₂ abruptly grows from ≈ 160 µs to 214 µs, Fig. 8(b). The different scaling laws for A_tot,max1(T_osc1) and A_tot,max2(T_osc1) strongly suggest that different mechanisms are responsible for generating the rarefaction waves leading to the first and second expansion of bubble clouds. They are most likely linked to laser-bubble-driven wall vibrations and liquid-mass oscillations in the small glass cuvette.

4. Discussion

4.1 Generation of secondary bubbles and their inhomogeneous distribution

Cavitation bubble formation relies on nucleation in the liquid to overcome the initial surface tension potential, which is large for small nuclei due to the 1/R scaling law. An extreme case is homogeneous nucleation without preexisting nuclei, which requires very large tensile stress to reach the kinetic spinodal. The tensile strength needs – 63 MPa at 168 °C for fs pulses with strong stress confinement conditions [20] and increases with increasing temperature [19]. For longer pulse durations without stress confinement, homogeneous nucleation requires the liquid to be heated up to 300 °C at 1 bar to reach phase explosion. By contrast, inhomogeneous nucleation with preexisting nuclei requires much less tensile strength for ultrashort pulses and reaches another extreme with boiling point (100 °C at 1 bar) for no stress confinement [52].

Rosselló et. al. observed nanobubble generation when he illuminated deionized water with a collimated ns pulse and passed a rarefaction wave through the pre-illuminated region [29]. The rarefaction wave was produced by reflecting a breakdown-induced shock wave at the liquid-air free surface. The intensity of the illuminating laser was about 3 × 10⁸ W/cm² and the fluence is about 1.6 J/cm². Using a similar concept and laser parameters, Jelenčič et al. provided experimental evidence that the nanobubbles were created by heating impurities in deionized water [30]. In both cases, nucleation was created by heating the impurities with the illuminating laser to produce long-lived tiny nanobubbles with a lifetime of > 1 ms and initial size ≥ 15 nm [29]. These nanobubbles are not visible in bright-field images but can be visualized using dark-field photography [53]. The rarefaction wave then drives the expansion of the “seeding” nanobubble nuclei to maximum radii of 10 µm ∼ 20 µm, which can be captured also by bright-field imaging [29,30].

In our study, bubble nuclei are created by heating impurities with a ns laser beam, as in the studies by Rosselló et.al. [29] and Jelenčič et. al. [30]. However, the rarefaction waves expanding the nuclei do not arise from the reflection of a breakdown shock wave at free surface, as in those papers. The images in Fig. 3(a) show that no secondary bubbles were observed when using the big cuvette although the distance from the free surface was the same as in the small cuvette. Therefore, we conclude that secondary cavitation in the small cuvette is not caused by tensile stress from shock wave reflection but rather by the wall vibrations. An estimate of the tensile stress amplitude arising from shock wave reflection under our experimental conditions shows that it is too weak to expand bubble nuclei to a size visible in our imaging system.

The distance between the laser focus and free surface in our experiments is 9 mm, and the traveling distance forth and back is 18 mm. According to the shock wave decay curve p_s(r) for the case of a laser bubble with similar size as the bubbles investigated in the present study, the amplitude of shock wave decays to ≈ 0.5 MPa at r = 18 mm [54]. Rosselló and Ohl reported in their study that the negative pressure after shock wave reflection is ≈ 65% of the positive peak pressure [29]. Thus, in our experiments a negative pressure of ≈ -0.325 MPa arising from the surface reflection of the breakdown shock wave is expected at the primary bubble location. The amplitude of this tensile stress transient is comparable to the amplitude of the rarefaction wave from wall vibrations but its duration is much shorter. Therefore, no visible bubbles beyond the resolution of our imaging system are produced by the passage of this short transient, while the rarefaction wave from wall vibrations can produce visible secondary cavitation. By contrast, Rosselló et al. and Jelenčič et. al. created tensile stress wave by producing optical breakdown very close to the water surface. This leads to tensile stress amplitudes of -5.2 MPa [29] and -15 MPa [30], which enabled to expand bubble nuclei even though the duration of the transients was below 100 ns.

The impurity radius is smaller than 110 nm in our study (filter size for the deionized water), which leads to an acoustic transmission time of less than 0.15 ns, much less than the pulse duration of 6 ns. Thus, there is hardly any stress confinement. Therefore, inhomogeneous nucleation by heating the impurities to a temperature ≥ 100 °C is very likely the mechanism leading to the formation of the initial nanobubble nuclei. For strong absorbers, inhomogeneous nucleation may occur even by applying 3.5 mW 80 MHz fs pulse series to nanometer-sized melanosomes [55].

The temperature increase at the impurities relies on their absorption cross section and the spatial distribution of light intensity, which in our case has a peak value at the focal center (> 3 × 10¹¹ W/cm²) and decreases rapidly along the axial and radial directions. This explains the inhomogeneous distribution of secondary bubbles in Fig. 7. With small pulse energies, the light intensity can create nanobubble nuclei only along the optical axis. With increasing pulse energy, nanobubble nuclei are gradually produced in the full cone angle. In all cases, no secondary bubbles were observed outside the cone angle of the laser beam due to the absence of laser light. This confirms that secondary bubbles cannot be produced by rarefaction waves alone.

For optical breakdown induced by ns pulses, the plasma shielding effect is substantial, which strongly absorbs the incident light and significantly reduces the transmitted light to the downstream side of the focus [48,56–58]. As a result, fewer and smaller bubble nuclei are produced on the downstream side, and secondary cavitation is weakened. Therefore, we observed an asymmetric distribution of secondary bubbles with a preference for the upstream direction (Fig. 7). A similar observation has been reported by Horvat et al. [46].

Following the above argument that correlates secondary cavitation with the spatial distribution of light intensity, we would expect secondary bubbles to be largest close to the focal spot and smaller towards the cuvette wall. However, we observed a zone without secondary bubbles in the vicinity of the laser bubble [Fig. 7(b)-(d) left column]. This phenomenon can be explained by the evolution of a ring of elevated pressure around the collapsing bubble, as will be explained in section 4.3.

The maximum radius of secondary bubbles evaluated from Fig. 7 ranges from tens of µm up to ≈ 100 µm, depending on pulse energy. This value is larger than the bubble size (10 µm ∼ 20 µm) observed by Rosselló et al.. Two reasons could explain the difference. Firstly, the light intensity in our study is much larger than the intensity of the collimated illuminating laser used by Rosselló et. al., which probably leads to a larger size of nanobubble nuclei. Secondly, the tensile component of the rarefaction wave that drives the expansion of the nanobubble nuclei in our study is much longer. According to our previous study [23], the rarefaction wave arises from the laser-bubble excited wall vibrations, and the tensile component has a duration of half the vibration period of the cuvette wall T_wall/2 = 58 µs. This value is more than 100 times longer than the period of tensile wave (< 0.5 µs) produced by reflecting a shock wave at a free surface. Although the amplitude of the tensile wave from the latter (≈ 50 bar) is much stronger than that in our study (several bars), the temporal integral of the tensile wave in our study may still be larger. Therefore, larger nanobubble nuclei and larger temporal integral of tensile wave are very likely the reasons leading to larger sizes of secondary bubbles in our study.

4.2 Origin of rarefaction waves in the partially confined small glass cuvette

In our previous study performed with a similar cuvette with partially confined surface [23], we hypothesized that rarefaction waves are very likely produced by wall vibrations, which are excited either directly by laser bubble or by liquid-mass oscillations. The hypothesis was supported by a good match between the experimentally measured dynamics of laser bubble [like Fig. 4(b)] with an extended Rayleigh-Plesset model that includes an external pressure term describing the rarefaction waves. In this study, we show evidence from the dynamics of secondary bubbles to strengthen this argument.

The secondary bubbles behave differently during the two expansion phases. Figure 9 shows two distinct scaling laws of A_tot,max1(T_osc1) and A_tot,max2(T_osc1): the former exhibits a monotonical increase following a growth function, whereas the latter shows an undulated pattern following a sinusoidal fitting function. T_osc1 represents the laser-bubble size as well as the driving frequency to the surroundings. As T_osc1 increases, the laser-bubble size increases, which leads to an increasing pressure in the liquid. This pressure feeds directly back on the laser bubble, leading to a reduced R_max value [Fig. 5(a)], and it exerts a force on the cuvette walls, leading to vibrations. A larger pressure amplitude results in stronger elastic deformation of the cuvette walls, followed by stronger tensile waves and larger secondary bubbles. However, the growth of the tensile stress is limited because the deformation of the cuvette walls is limited by geometrical factors and the stress-strain behavior of the material. Therefore, the A_tot,max1(T_osc1) curve can be fitted by a growth function that reaches a saturation level at large T_osc1. Coalescence of secondary bubbles at large pulse energies [Fig. 7(d) left] can also contribute to the saturation of A_tot,max1(T_osc1) at large T_osc1.

For liquid-mass oscillations, the situation is different. Due to the partial confinement of the free surface, the expansion of the laser bubble imparts an impulse to the surrounding liquid, which accelerates it towards the cuvette corners, where the liquid surface is not confined by the transducer. The mechanical energy of the laser bubble is partly transformed first into kinetic and then into potential energy of the liquid mass in the cuvette corners. The inertial movement of the liquid-mass is counteracted by gravity and damped by friction along transducer and cuvette walls. The oscillation period of the liquid-mass oscillator depends on the geometry of the cuvette and the mass of liquid, whereas the oscillation amplitude is correlated with the laser bubble size. With increasing T_osc1, the liquid-mass oscillations become initially stronger but then weaker because the phase between laser-bubble oscillation and liquid-mass oscillation shifts. This explains why the A_tot,max2(T_osc1) curve in Fig. 9 can be fitted by a sinusoidal function, which has a peak value at T_osc1 ≈ 160 µs, very close to the liquid-mass oscillation period T_lm = 160 µs. The secondary bubbles are maximally expanded when the laser-bubble and the liquid-mass oscillation are in resonance. For T_osc1 > 160 µs, the size of secondary bubbles decreases because the exciting frequency (T_osc1) goes off-resonance.

We found through an additional experiment that the liquid-mass oscillation alone cannot produce rarefaction waves strong enough to generate secondary bubbles. We produced laser bubbles in a fenestrated metal cuvette with same inner size (12 mm width and 45 mm height) but several mm thick walls that suppressed wall vibrations. Bubbles were produced 18 mm above the cuvette bottom and 9 mm below the surface, with the same transducer just touching the water surface. However, we could not detect secondary bubble formation using the same pulse energies as with the thin-walled glass cuvette (data not shown). Thus, the involvement of wall vibrations is essential for secondary bubble generation.

4.3 Interplay between the laser bubble and secondary cavitation bubbles

Since nuclei for secondary cavitation bubbles are created by laser-heating of liquid impurities, one may expect many large secondary bubbles close to the focal spot, where the light intensity is largest. However, we observed a broad zone (up to 350 µm) void of secondary bubbles in the vicinity of the laser bubble [Fig. 7(b)-(d) left column]. This phenomenon can be explained by the evolution of a ring of elevated pressure around the collapsing bubble that counteracts the rarefaction wave from the wall vibrations. During the collapse of a spherical laser bubble, not only the bubble interior is compressed but also the surrounding liquid around the collapsing bubble. Liang et al. simulated the pressure distribution for a spherical laser bubble of R_max = 35 µm at collapse and rebound [59] and found that there is a compressed shell in the liquid surrounding the collapsing bubble. The pressure in this ring is as high as 10 MPa close to the bubble shortly before collapse and drops to ambient pressure (0.1 MPa) about 200 µm away from the bubble wall. For larger sub-mm-sized laser bubbles as in our study, the spatial extent of the zone with elevated pressure in the liquid around the laser bubble is significantly larger. We would expect a drop to about 3 bar at 350 µm because the tensile stress arising from the wall vibrations from our previous study has a value ≈ -3 bar for R_max = 600 µm [23]. The elevated pressure evolving during bubble collapse counteracts the tensile wave arising from the wall vibrations and inhibits the expansion of nanobubble nuclei. Therefore, no secondary bubbles are produced in the immediate vicinity of the laser bubble up to a distance, where the amplitude of the rarefaction wave is larger than the pressure elevation produced by the collapsing bubble.

During the second expansion of the secondary bubble clouds [Fig. 4(c) and Fig. 7, right column], not only the secondary bubbles but also the laser bubble oscillation are governed by the rarefaction wave [Fig. 4(b)]. Since the laser bubble expansion by the rarefaction wave produces no elevated pressure in the liquid, it will not inhibit the expansion of surrounding nanobubble nuclei in its vicinity. Therefore, secondary bubbles can now appear also in the vicinity of the laser bubble, different from the first expansion cycle.

For large pulse energies, the laser bubble migrates during its first collapse toward the secondary bubbles on the upstream side [Fig. 7, right column, and Figs. S3-S5]. Obviously, the oscillating cloud of secondary bubbles creates a pressure gradient favoring the migration of the laser bubble. Such migration is often observed in bubble pair interactions. For example, Tinne et al. and Han et al. observed that spatially separated bubble pair that are simultaneously produced by two pulses of different energies migrates toward each other [60,61]. A similar phenomenon was observed by Zhang et al. in spark-induced bubble pairs [62].

5. Conclusions

We investigated secondary cavitation bubble dynamics during the laser-bubble formation in a small container with a partially confined free surface and elastic thin walls. The dynamics of laser bubble and secondary bubble clouds were recorded by high-speed photography. Acoustic and light scattering measurements were used to detect the oscillation times of the laser bubble. The appearance of secondary bubbles coincides with the prolongation of the collapse phase as well as the re-oscillations of the laser bubble.

Nucleation of secondary bubbles are created through heating of impurities by the laser light, and these nuclei are then expanded to tens of µm or even ≈ 100 µm by rarefaction waves, which have a strength of a few bars and last tens of microseconds. The rarefaction waves originate from vibrations of the thin elastic cuvette walls, which are excited by the oscillations of the laser bubble and through bubble-excited liquid-mass oscillations in the cuvette corners around the piston-like transducer. The amplitude of the rarefaction wave is strong during the collapse phase of the laser bubble, when the cuvette wall moves still outward while the bubble wall already moves inward. Secondary bubbles are maximally expanded when liquid-mass oscillator and laser bubble are in resonance.

Secondary cavitation bubbles were observed mainly on upstream side of the focus because plasma shielding lowered the laser light intensity on the downstream side, which results in a smaller density of secondary bubble nuclei. Secondary bubbles were absent in the immediate vicinity of the laser bubble because the inward-directed radial flow during collapse produces a pressure ring that counteracts the rarefaction wave from wall vibrations.

Our findings are relevant for applications involving laser-induced cavitation in partially confined spaces that are surrounded with elastic thin walls, such as intraluminal laser surgery [15,16,18] and laser-based cell sorting in microfluidics [11,12]. Understanding the dynamics of secondary bubbles and laser bubbles, and their mutual interactions can be useful to enhance the efficiency in processes such as rapid mixing [13] and micro-pumping [14] in microfluidic channels.