Expansion Dynamics After Laser-Induced Cavitation in Liquid Tin Microdroplets
Dmitry Kurilovich, Tiago de Faria Pinto, Francesco Torretti, Ruben Schupp, Joris Scheers, Aneta S. Stodolna, Hanneke Gelderblom, Kjeld S. E. Eikema, Stefan Witte, Wim Ubachs, Ronnie Hoekstra, Oscar O. Versolato
EExpansion Dynamics After Laser-Induced Cavitation in Liquid Tin Microdroplets
Dmitry Kurilovich,
1, 2
Tiago de Faria Pinto,
1, 2
Francesco Torretti,
1, 2
Ruben Schupp, Joris Scheers,
1, 2
Aneta S. Stodolna,
1, 2
Hanneke Gelderblom, Kjeld S. E. Eikema,
1, 2
Stefan Witte,
1, 2
Wim Ubachs,
1, 2
Ronnie Hoekstra,
1, 4 and Oscar O. Versolato ∗ Advanced Research Center for Nanolithography, Science Park 110, 1098 XG Amsterdam, The Netherlands Department of Physics and Astronomy, and LaserLaB, Vrije Universiteit Amsterdam,De Boelelaan 1081, 1081 HV Amsterdam, The Netherlands Department of Applied Physics, Eindhoven University of Technology,P.O. Box 513, 5600 MB Eindhoven, The Netherlands Zernike Institute for Advanced Materials, University of Groningen,Nijenborgh 4, 9747 AG Groningen, The Netherlands (Dated: May 21, 2018)The cavitation-driven expansion dynamics of liquid tin microdroplets is investigated, set in motionby the ablative impact of a 15-ps laser pulse. We combine high-resolution stroboscopic shadowgraphywith an intuitive fluid dynamic model that includes the onset of fragmentation, and find goodagreement between model and experimental data for two different droplet sizes over a wide rangeof laser pulse energies. The dependence of the initial expansion velocity on these experimentalparameters is heuristically captured in a single power law. Further, the obtained late-time massdistributions are shown to be governed by a single parameter. These studies are performed underconditions relevant for plasma light sources for extreme-ultraviolet nanolithography.
I. INTRODUCTION
Intense, short-pulse laser radiation can produce strongshock waves in liquids leading in some spectacularcases to explosive cavitation and violent spallation ofthe material [1–5]. Such dramatic physical phenomenacan readily find application, a very recent example ofwhich is found in the field of nanolithography wheremicrodroplets of liquid tin are used to create extremeultraviolet (EUV) light [6–8]. These tin droplets,typically several 10 µ m in diameter, serve as targets fora high-energy, ns-pulse laser, creating a laser-producedplasma (LPP). Line emission from highly charged tinions in the LPP provides the required EUV light. Cur-rently, a dual-laser-pulse sequence is employed [8]. In afirst step, a ns-laser prepulse is used to carefully shapethe droplet into a thin sheet that is considered to beadvantageous for EUV production with a second, muchmore energetic, main pulse. Recent developments [8, 9],however, produced tentative but tantalizing evidence forsignificantly improved source performance when replac-ing the ns-prepulse with a ps-prepulse laser to produce ashock-wave-induced explosive fragmentation. Althoughsome notable progress was made very recently [3–5, 10],this process requires further investigation.This paper advances the understanding of the afore-mentioned systems by providing an experimental andtheoretical study of the late-time dynamics of thedeformation of free-falling tin microdroplets. Beinginitially spherical, the droplets are subjected by strongshock waves generated by ps laser pulse impact, givingrise to cavitation [3–5, 10]. This centralized cavitation ∗ [email protected] explosively propels the liquid to very high ( ∼
100 m/s)radial velocities, producing a rapidly thinning liquid-tinshell (see Fig. 1). The initial spherical symmetry of thesystem is broken when the intensity of the shock waveexceeds a certain threshold and dramatic spallation isobserved on the side of the droplet facing away fromthe laser impact zone. We focus our studies on the richphysics of the dynamics set in motion by the centralcavitation.We develop a model description for the time evolutionof a stretching spherical shell, including the onset of frag-mentation. Using stroboscopic shadowgraphic imaging,this model is experimentally validated over a wide rangeof laser pulse energies, 15 ps in duration, and for twodroplet sizes. The dependence of the initial expansionvelocity on these experimental parameters is heuristicallycaptured in a single power law. The late-time mass distri-butions, experimentally obtained from front-view shad-owgraphy, are furthermore shown to be governed by asingle parameter.
II. EXPERIMENTAL METHODS
Our experimental setup has previously been describedin detail [11]. For clarity, the most important characteris-tics, as well as experimental upgrades, are discussed here.A droplet generator is operated in a vacuum chamber(10 − mbar) and held at constant temperature of 260 ◦ C,well above the melting point of tin. The nozzle producesan on-axis multi-kHz train of 15 or 23 µ m radius dropletsof 99.995% purity tin, with density ρ = 7 g/cm andsurface tension σ = 0 .
532 N/m. The droplets relax toa spherical shape before they pass through a horizontallight sheet produced from a helium-neon laser. The lightscattered by the droplets is detected by a photomulti- a r X i v : . [ phy s i c s . f l u - dyn ] M a y FIG. 1. Stroboscopic shadowgraph images of expanding tinmicrodroplets (23 µ m initial radius) taken at different timedelays for three different laser pulse energies (or Weber num-bers, see Sec. III A) at a pulse length of 15 ps, as seen fromtwo viewing angles (90 ◦ side view and 30 ◦ front view). Laserimpacts from the left; images are cropped and centered indi-vidually to improve visibility. A 500- µ m-length scale bar isprovided in the left-lower corner. plier tube, which signal is reduced in frequency to 10 Hzto trigger a Nd:YAG laser system. This laser systemproduces tunable ps pulses of 1064 nm wavelength lightas described in more detail in Ref. [12]. The laser pulseenergy is varied between 0.5 and 5 mJ employing a suit-able combination of a waveplate and polarizer; its pulselength is kept constant at approximately 15 ps for theexperiments described here. The laser beam is focuseddown to a 100 µ m full-width-at-half-maximum (FWHM)diameter Gaussian spot. In order to maintain cylindricalsymmetry, the laser light has circular polarization. Anaccurately timed laser pulse provides a radially-centeredinteraction with a falling droplet that occurs in a repro-ducible manner. Only a small fraction of the dropletsinteract with a laser pulse due to the mismatch in rep-etition rates of the droplet generator (multi-kHz) andlaser (10 Hz).Two shadowgraph imaging systems capture the dy-namics of the expanding droplets. These systems arebased on a single pulse from a broadband 560 ± µ m resolution. One of these microscopes is aligned or-thogonally to the laser beam to obtain side-view images;the other one is at 30 ◦ angle to the beam direction for a(tilted) front view. Both microscopes are equipped withbandpass filters to suppress the plasma radiation. Theobtained images are used to track the size, shape, andvelocity of the droplet expansion employing an image-processing algorithm. Stroboscopic time series of differ-ent droplets are constructed by triggering the shadowg-raphy systems once per drive laser shot with increasingdelay, typically with 50–100 ns time steps [13]. III. RESULTS AND INTERPRETATION
The response of a tin droplet to laser pulse impact isshown in Fig. 1 for three laser pulse energies. A qualita-tive description of the relevant physical processes leadingto cavitation and spallation was recently given in Ref. [4].We summarize the crucial steps in the following, com-bining it with our experimental observations. The laserpulse impact ablates a thin (less than 1 µ m) layer of tin.At the lower pulse energies (see the 0.7 or 1.5 mJ casespresented in Fig. 1), this ablated mass is clearly visibleand seen to move away from the droplet in the direc-tion opposite to the laser light. The remarkably sharpouter boundary of this ablated mass may be explained bythe existence of an inhomogeneous two-phase, gas-liquidmixture of low average density but approaching liquiddensity in the vicinity of the ablation front [14, 15]. Incontrast to the ns laser pulse impact [11], the resultingdroplet propulsion is limited here.The ablation pulse gives rise to a pressure wave start-ing from the laser-impacted region of the droplet. Thispressure is applied on a short, ps time scale that is sev-eral orders of magnitude shorter than the time scale onwhich pressure waves travel through the droplet.As a consequence, the liquid is locally compressed andpressure waves travel through the droplet [16]. Converg-ing pressure ”shock” waves superimpose in the center ofthe droplet where they cause tensile stress. Once thistensile stress is above the yield strength of the liquid itruptures and a thin shell is formed. Theoretical modeling[4] has shown that there is a high sensitivity of both therate of expansion and the morphology of the deformeddroplet to the details of the, poorly known, metastableequation-of-state (EOS) of tin in the region of the liquid-vapor phase transition and to the parameters of the crit-ical point. Thus, precise theory predictions of the post-cavitation dynamics of liquid tin microdroplets cannotyet be made and, instead, measurements as presented inthis work may provide a sensitive instrument for probingthe EOS of liquid metals [4].After passing through the droplet center, the shockwave reaches the back side of the droplet (i.e. the sidefacing away from the laser) and reflects from this free in-terface where it can again rupture the liquid and give riseto spallation that rapidly propels, at several 100 m/s, asmall mass fraction of the droplet along the laser beampropagation direction (see Fig. 1) [4, 5]. The fluid dy-namic description of this spallation is very rich and willbe left for future work. However, it is noteworthy thatthere is a regime where a strong spallation occurs, but ismitigated under capillary action (compare the t = 1 and2 µ s shadowgraphs for the 1.5 mJ case in Fig. 1).We note that the higher-energy cases in Fig. 1 alsohint at the existence of a hole on the side of the dropletfacing the laser, which may possibly be attributed toablation-thinning of the droplet shell. In combinationwith the spallation, a ”tunnel” is thus created (see thefront view shadowgraphs in Fig. 1), that may later forma doughnut-type mass distribution. Another scenario,when fragmentation following the collapse of the shellresults in high-speed jetting, corresponding to the low-est laser pulse energies, is reported on in Refs. [10, 17, 18].In the following, we focus on the late-time dynamicsset in motion by the central cavitation. Firstly, a basicmodel for the time evolution of the intact liquid shellis presented. Secondly, we study the time scale atwhich first holes in this shell become visible, after whichfull fragmentation of the shell rapidly sets in. Thisresults, eventually, in a late-time mass distribution, theunderstanding of which is of particular relevance forproducing EUV radiation at high conversion efficiencies. A. Time evolution of the shell
The shadowgraphs such as those presented in Fig. 1clearly show a cylindrically symmetric expanding shell,the radius R ( t ) of which is tracked for each time stepby measuring the maximal transverse (vertical in Fig. 1)size of this feature from the side-view shadowgraphs.By fitting a linear function to the first few R ( t ) datapoints after the laser impact, we obtain values for theinitial droplet expansion velocity ˙ R ( t =0). Figure 2(a)shows a monotonic increase of this radial expansionvelocity ˙ R ( t =0) as a function of laser pulse energy.Using a heuristic argument, we collapse all obtaineddata onto a single curve, scaling the obtained velocitieswith the initial droplet radius R (see Fig. 2(b)). Toarrive at this scaling, we hypothesize that the magnitudeof the induced shock wave, and its related cavitationevent, scales with the laser-facing surface area, ∝ R (the droplet is much smaller than the laser spot size).Meanwhile, the expansion is impeded by the dropletmass ∝ R − . By neglecting the possible differences inthe dissipation of the shock wave, one thus arrives at theaforementioned simple scaling ˙ R ( t =0) ∝ R − . As seenfrom Fig. 2(b), such a rescaling causes all data to fallon a single curve that appears to be represented quitewell by a power-law function ˙ R ( t =0) R = A × E α withfit parameters A = 9.4 ± . / s and α = 0.46 ± . -3 -3 R ( t = )( m / s ) (b)(a) Radius R (cid:181)m 23 (cid:181)m fit R ( t = ) R ( m / s ) E (mJ)
FIG. 2. (a) Initial radial expansion velocity ˙ R ( t =0) of tin mi-crodroplets as a function of total laser pulse energy E . Yel-low diamonds are for R = 15 µ m and blue squares are for R = 23 µ m. The error bars represent 10% uncertainty on thevelocity measurements and 20% uncertainty on the energymeasurements. (b) The same data, rescaled by multiplyingby the initial droplet radius R . The solid red line is obtainedby fitting a power-law function to the concatenated data. parameters cannot be straightforwardly predicted fromavailable theory, given their sensitivity to the EOS andthe details of laser-matter interaction.Next, to describe the time evolution of the liquiddroplets in Fig. 1, we hypothesize that a small ( ∼ µ m),high-pressure ( ∼ kbar) cavitation bubble expands the liq-uid quickly into a thin shell. Thus, a spherically symmet-ric shell expands at a certain initial velocity ˙ R ( t =0), withthe cavitation pressure having done its thermodynamicwork effectively at time zero. Given that the dropletexpands into the vacuum, the only limitation to its ex-pansion is surface tension.The liquid tin shell has thickness h ( t ) which de-creases quadratically with time as mass conservation im-plies, in the limit of a thin shell ( R ( t ) (cid:29) h ( t )), that h ( t ) = R / (3 R ( t ) ) ≈ R / (3 t ˙ R ( t =0) ). The expan-sion of the shell can be described by the Rayleigh-Plessetequation, which in the limit of a thin shell reads [22]: p ( R − h ) − p ( R ) ≈ ρh ¨ R , where p is pressure. Imposingdynamic boundary conditions at the two liquid-vapor in-terfaces p ( R − h ) and p ( R ), and noting that the only forceacting on the shell is the Laplace pressure, the net valueof which is given by p = 4 σ/R ( t ) (in contrast to Ref. [22],where this contribution was negligible), we find¨ R ( t ) = − pρh ( t ) = − σρh ( t ) R ( t ) = − σR ( t ) ρR . (1)From the above expression, using the boundary condi-tions of a given ˙ R ( t =0) and initial radius R , one obtains R ( t ) R = cos( πt/τ c ) + (cid:114) We12 sin( πt/τ c ) , (2)introducing the capillary time scale here as τ c = π (cid:112) ρR / σ and the relevant Weber numberas We = ρR ˙ R ( t =0) /σ . The above solution is identicalin form to that obtained previously for a disk-typeexpansion of a droplet following ns-laser pulse im-pact [11, 23].Next, we apply the theory developed above withzero fit parameters taking as an input the experimentalvalues for ˙ R ( t =0). The results are presented in Fig. 3(b),where the experimentally obtained radius R ( t ) (seeFig. 3a) is rendered dimensionless by dividing by R (and subtracting the unity value offset), and plottedas a function of dimensionless time t/τ c . Given thesimplicity of our arguments, we find excellent agreementbetween model and experiment. The observed late-time”overshoot” that is apparent in the experimental datashown, with theory predicting a more rapid retraction ofthe shell, can be understood by considering the processof hole formation and ensuing fragmentation, whichstrongly reduces the restoring forces. B. Hole opening time
The expanding shell is subjected to Rayleigh-Taylor in-stabilities (RTI), here as instabilities of the liquid-vaporinterface driven by the radial acceleration and deceler-ation of the liquid sheet [22]. Surface tension providesmode selection of the RTI [22, 24–26]. This phenomenonwill lead to hole formation after which rapid hole openingand merging will lead to full fragmentation of the shell.In our system, there are two acceleration mechanismsthat act along the surface normal and that thus will con-tribute to RTI growth. Firstly, the liquid is strongly ac-celerated by the cavitation pressure. Secondly, after theinitial fast expansion, the shell much more slowly decel-erates under the influence of surface tension. Holes willform at the shell piercing time t ∗ , when the size of a grow-ing instability is of the order of the shell thickness [22].In the available literature (e.g., see Refs. [22, 26]), quitegenerally a scaling relation is found of the type t ∗ τ c ∝ We − β , (3)resulting in a rough scaling for the corresponding desta-bilization radius [22] R ( t ∗ ) R − ∝ We / − β . (4)Here we have, for simplicity, linearized the expansionrate to a constant ˙ R ∝ We / and in Eq. (3) droppedthe usual weak scaling term ( η /R ) β , where η isthe initial amplitude of RTI. The positive power β is R ( t ) FIG. 3. (a) Radius R ( t ) of shell driven by cavitation in R = [15] 23 µ m droplets for similar Weber numbers. [Open]black squares: We = [304] 368; [open] brown circles: We =[705] 754; [open] green triangles We = [1923] 1778. The in-set (500 µ m × µ m) represents a side-view image of theWe = 754 case at 2.5 µ s. (b) Rescaled, dimensionless radiusas a function of dimensionless time t/τ c . The solid curves rep-resent the model predictions (see Eq. (2)) for the 23 µ m initialradius droplet; the dashed curves show the same but for thesmaller, 15 µ m droplet. The solid gray line depicts the desta-bilization radius R ( t ∗ ) as obtained from the fit of Eq. (3) tothe data (see Fig. 4), applied to Eq. (2). The data are grayedout beyond this line. typically smaller than unity [22, 26, 27]. Obtaining atheory value for this power is complicated because ofthe two competing mechanisms driving RTI. We takeinstead the rather general form of Eq. (3) and let theexperiment provide the relevant value for β and forthe proportionality constant. In the experiments, ahole becomes clearly visible in side view shadowgraphyonly when another hole simultaneously appears onthe opposite side. This naturally introduces a slightdetection bias. However, it is possible to reliably andreproducibly obtain an estimate for t ∗ based on opticalinspection of the experimental data supported by thefact that for times t > t ∗ (where we take 100 ns timesteps) the shell becomes permeated with holes. Weobserve a monotonically decreasing value for t ∗ with
500 1000 1500 2000 25000.20.250.30.350.4
Radius R (cid:181)m 23 (cid:181)m fit t * / c We R ( t * ) / R We FIG. 4. Dimensionless time of first hole opening t ∗ /τ c as afunction of Weber number for two droplet sizes: 15 µ m (yel-low diamonds) and 23 µ m initial radius (blue squares). ForWe (cid:46) β = 0.34 (see main text). increasing Weber number as expected from Eq. (3).Furthermore, t ∗ is seen to decrease with droplet sizefor similar Weber number. Rescaling t ∗ → t ∗ /τ c inFig. 4 indeed collapses the data for the two differentdroplet sizes onto a single curve. The resulting fit valuesof the power law Eq. (3) are given by β = 0.34(4) and3.0(8) for the proportionality constant. As we clearlyfind β < R ( t ∗ ) increaseswith Weber number, and thus with laser energy. Theobtained fit values will serve as input for predicting thelate-time mass distribution produced by the laser impact. C. Late-time mass distribution
Having achieved good agreement between experimentand model, we now summarize our findings in Fig. 5 tofacilitate a more direct, industrial application of ourfindings. The maximum amplitude of an intact retract-ing shell is given by maximizing Eq. (2). The resultingdashed black curve in Fig. 5 is close to the well-knownscaling ∼√ We (as was pointed out in Ref. [28]). Themaximum obtainable shell size without any holes (i.e.the destabilization radius) is obtained by insertingEq. (3) into Eq. (2) as in Fig. 3. The result is shown as asolid black line in Fig. 5. Once this line is crossed in theexpansion phase, the rapid breakup and fragmentationof the shell, associated with large Weber numbers here,precludes capillary collapse and the droplet target fragments ballistically expand. For the low Weber num-bers shown the shell collapses and jetting may ensue [10].In Fig. 5 we have also indicated ranges of Weber num-bers that we associate with different spallation regimes.For Weber numbers below We ≈ (cid:46) We (cid:46) (cid:38) (cid:38) R ( t =0).From an application perspective, it is interesting tostudy the ”final” late-time fragment mass distributionthat might serve as a target for a main laser pulse in anactual industrial EUV-source based on tin plasma. In thefollowing analysis, we present some brief guidelines basedon the here presented data but keep the full analysis ofthe relevant fragmentation process (including fragmentsize distributions) for a later, dedicated work. We findthat the final mass distribution changes very dramati-cally over a relatively small range of Weber numbers ascan be seen from the shadowgraphs presented in Fig. 6(a)as well as from their angularly averaged, radial projec-tions shown in Fig. 6(b). These projections can loosely beinterpreted as a column-density mass distribution alongthe line-of-sight of the backlight illumination. The pro-jections are not corrected for the small parallax angle orfor the limited depth of focus. Still, we expect to tracka large fraction of the total droplet mass. Similar massdistributions are found for 15 and 23 µ m droplets at com-parable Weber numbers (see Fig. 6). spall breakoutspall retractionno spall b a ll i s t i c e x p a n s i o n o f f r a g m e n t s R / R We h o l e f o r m a t i o n a n d b r e a k - u p collapse FIG. 5. Phase diagram depicting the maximum obtainableradius as a function of Weber number. White arrows illustratethe various spallation regimes (see main text). The dashedline is obtained by maximizing Eq. (2). The solid line showsthe destabilization radius as obtained from inserting Eq. (3)into Eq.( 2); the area under this line is obtainable withoutholes occurring. R = µ m R = µ m µ m µ m (a) (b) M a ss d e n s i t y ( a r b . un i t s ) We =
544 We =
754 We =
705 We = µ s 8.9 µ s10.9 µ s2.5 µ s 4.9 µ s FIG. 6. (a) Late-time front (30 ◦ ) view shadowgraphs of ex-panded and deformed droplets of different droplet sizes rep-resenting cases of similar Weber numbers. The rectangularscale grid in the top-left corner visualizes the effect of the30 ◦ parallax angle. (b) Inverted, angularly averaged, radialprojections of the obtained gray scale of corresponding im-ages from (a), which gives a qualitative measure of the radialcolumn-density-mass distribution. The dashed red line corre-sponds to the data from R = 15 µ m droplets; the solid blackline is for R = 23 µ m. For We = 544 we observe that the mass distributionhas a maximum in the center of the image, as is to beexpected from a collapse. For just very slightly largerWeber numbers, we find instead a toroidal profile that be-comes more and more pronounced for larger Weber num-bers. We attribute this observation to the RTI-drivenbreakup of the ”tunnel” wall (thus preventing collapse)as well as by a pronounced spallation that modifies themass distribution, as was also noted in Ref. [4].The here established and quantified sensitivity ofthe final state mass distribution to the Weber numbercan readily find application. As an example, we notethat a typical target size used in the industry is inthe order of a few hundred micrometers [8], several µ safter prepulse impact. In our experiments, a relevant ∼ µ m diameter target is, for example, obtained froma R = 15 µ m initial droplet size after expanding for2.5 µ s, see Fig. 6(a) for We = 1923. In this case, the mainpulse would not find a significant mass fraction in itsfocus. Thus, it may well be opportune to choose a lowerWeber number, by tuning down the laser energy, andobtain a more homogeneous mass distribution whichcould improve the conversion efficiency of drive laserlight into useful EUV radiation by, e.g., improving the absorption of drive laser light. IV. CONCLUSIONS
We present an analysis of the cavitation-driven expan-sion dynamics of liquid tin microdroplets that is set inmotion by the ablative impact of a 15-ps laser pulse.High-resolution stroboscopic shadowgraphy of the ex-panding tin shells is combined with an intuitive fluid dy-namic model that includes the time and size at which theonset of fragmentation becomes apparent. This modelwill aid follow-up studies of the fragmentation pathways.Good agreement between model and experimental datais found for two different droplet sizes over a wide rangeof laser pulse energies. The dependence of the initial ex-pansion velocity of the liquid shell on these experimen-tal parameters is heuristically captured, for applicationpurposes, in a single power law. A summary phase dia-gram of the expansion dynamics is presented. It coversregimes with and without spallation, as well as a transi-tion regime where the spalled material is retracted undersurface tension. This transition regime enables findingconditions for a maximum shell expansion velocity withstrongly suppressed forward-moving debris. This phasediagram facilitates a more direct, industrial applicationof our findings.Further, the experimentally obtained late-time massdistributions are shown to be governed by a singleparameter, the Weber number. These studies are ofparticular relevance for plasma sources of extremeultraviolet light for nanolithography. In such plasmasources, the tin mass distributions obtained uponcavitation-driven shell fragmentation, as studied in thiswork, are shown to be promising targets for efficientlaser coupling [8, 9].
ACKNOWLEDGMENTS
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