# Expansion Dynamics After Laser-Induced Cavitation in Liquid Tin
Microdroplets

Dmitry Kurilovich  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  Tiago de Faria Pinto  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  Francesco
Torretti  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  Ruben Schupp  Advanced Research Center for Nanolithography,
Science Park 110, 1098 XG Amsterdam, The Netherlands  Joris Scheers  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
Aneta S. Stodolna  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  Hanneke Gelderblom  Department of Applied Physics,
Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The
Netherlands  Kjeld S. E. Eikema  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  Stefan Witte  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  Wim Ubachs
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  Ronnie Hoekstra  Advanced Research Center for Nanolithography,
Science Park 110, 1098 XG Amsterdam, The Netherlands  Zernike Institute for
Advanced Materials, University of Groningen, Nijenborgh 4, 9747 AG Groningen,
The Netherlands  Oscar O. Versolato<EMAIL_ADDRESS>Advanced Research
Center for Nanolithography, Science Park 110, 1098 XG Amsterdam, The
Netherlands

###### Abstract

The cavitation-driven expansion dynamics of liquid tin microdroplets is
investigated, set in motion by the ablative impact of a 15-ps laser pulse. We
combine high-resolution stroboscopic shadowgraphy with an intuitive fluid
dynamic model that includes the onset of fragmentation, and find good
agreement between model and experimental data for two different droplet sizes
over a wide range of laser pulse energies. The dependence of the initial
expansion velocity on these experimental parameters is heuristically captured
in a single power law. Further, the obtained late-time mass distributions are
shown to be governed by a single parameter. These studies are performed under
conditions relevant for plasma light sources for extreme-ultraviolet
nanolithography.

##  I Introduction

Intense, short-pulse laser radiation can produce strong shock waves in liquids
leading in some spectacular cases to explosive cavitation and violent
spallation of the material Stan  _et al._ (2016a, b); Vinokhodov  _et al._
(2016); Basko  _et al._ (2017); Krivokorytov  _et al._ (2017). Such dramatic
physical phenomena can readily find application, a very recent example of
which is found in the field of nanolithography where microdroplets of liquid
tin are used to create extreme ultraviolet (EUV) light Banine  _et al._
(2011); Pirati  _et al._ (2017); Fomenkov  _et al._ (2017). These tin
droplets, typically several 10 $\mu$m in diameter, serve as targets for a
high-energy, ns-pulse laser, creating a laser-produced plasma (LPP). Line
emission from highly charged tin ions in the LPP provides the required EUV
light. Currently, a dual-laser-pulse sequence is employed Fomenkov  _et al._
(2017). In a first step, a ns-laser prepulse is used to carefully shape the
droplet into a thin sheet that is considered to be advantageous for EUV
production with a second, much more energetic, main pulse . Recent
developments Fomenkov  _et al._ (2017); Kawasuji  _et al._ (2017), however,
produced tentative but tantalizing evidence for significantly improved source
performance when replacing the ns-prepulse with a ps-prepulse laser to produce
a shock-wave-induced explosive fragmentation. Although some notable progress
was made very recently Vinokhodov  _et al._ (2016); Basko  _et al._ (2017);
Krivokorytov  _et al._ (2017, 2018), this process requires further
investigation.

This paper advances the understanding of the aforementioned systems by
providing an experimental and theoretical study of the late-time dynamics of
the deformation of free-falling tin microdroplets. Being initially spherical,
the droplets are subjected by strong shock waves generated by ps laser pulse
impact, giving rise to cavitation Vinokhodov  _et al._ (2016); Basko  _et al._
(2017); Krivokorytov  _et al._ (2017, 2018). This centralized cavitation
explosively propels the liquid to very high ($\sim$100 m/s) radial velocities,
producing a rapidly thinning liquid-tin shell (see Fig. 1). The initial
spherical symmetry of the system is broken when the intensity of the shock
wave exceeds a certain threshold and dramatic spallation is observed on the
side of the droplet facing away from the laser impact zone. We focus our
studies on the rich physics of the dynamics set in motion by the central
cavitation.

We develop a model description for the time evolution of a stretching
spherical shell, including the onset of fragmentation. Using stroboscopic
shadowgraphic imaging, this model is experimentally validated over a wide
range of laser pulse energies, 15 ps in duration, and for two droplet sizes.
The dependence of the initial expansion velocity on these experimental
parameters is heuristically captured in a single power law. The late-time mass
distributions, experimentally obtained from front-view shadowgraphy, are
furthermore shown to be governed by a single parameter.

Figure 1:  Stroboscopic shadowgraph images of expanding tin microdroplets (23
$\mu$m initial radius) taken at different time delays for three different
laser pulse energies (or Weber numbers, see Sec. III.1) at a pulse length of
15 ps, as seen from two viewing angles (90∘ side view and 30∘ front view).
Laser impacts from the left; images are cropped and centered individually to
improve visibility. A 500-$\mu$m-length scale bar is provided in the left-
lower corner.

##  II Experimental methods

Our experimental setup has previously been described in detail Kurilovich  _et
al._ (2016). For clarity, the most important characteristics, as well as
experimental upgrades, are discussed here. A droplet generator is operated in
a vacuum chamber ($10^{-7}$ mbar) and held at constant temperature of 260∘ C,
well above the melting point of tin. The nozzle produces an on-axis multi-kHz
train of 15 or 23 $\mu$m radius droplets of 99.995% purity tin, with density
$\rho=7\,$g/cm3 and surface tension $\sigma=0.532$ N/m. The droplets relax to
a spherical shape before they pass through a horizontal light sheet produced
from a helium-neon laser. The light scattered by the droplets is detected by a
photomultiplier tube, which signal is reduced in frequency to 10 Hz to trigger
a Nd:YAG laser system. This laser system produces tunable ps pulses of 1064 nm
wavelength light as described in more detail in Ref. Stodolna  _et al._
(2018). The laser pulse energy is varied between 0.5 and 5 mJ employing a
suitable combination of a waveplate and polarizer; its pulse length is kept
constant at approximately 15 ps for the experiments described here. The laser
beam is focused down to a 100 $\mu$m full-width-at-half-maximum (FWHM)
diameter Gaussian spot. In order to maintain cylindrical symmetry, the laser
light has circular polarization. An accurately timed laser pulse provides a
radially-centered interaction with a falling droplet that occurs in a
reproducible manner. Only a small fraction of the droplets interact with a
laser pulse due to the mismatch in repetition rates of the droplet generator
(multi-kHz) and laser (10 Hz).

Two shadowgraph imaging systems capture the dynamics of the expanding
droplets. These systems are based on a single pulse from a broadband 560$\pm
2$ nm wavelength 5-ns-pulse dye laser. Combined with two long-distance
microscopes coupled with CCD cameras, this laser source provides back-lighting
that is carefully rendered incoherent both temporally and spatially, with a 4
$\mu$m resolution. One of these microscopes is aligned orthogonally 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 with
bandpass filters to suppress the plasma radiation. The obtained images are
used to track the size, shape, and velocity of the droplet expansion employing
an image-processing algorithm. Stroboscopic time series of different droplets
are constructed by triggering the shadowgraphy systems once per drive laser
shot with increasing delay, typically with 50–100 ns time steps SM .

##  III Results and interpretation

The response of a tin droplet to laser pulse impact is shown in Fig. 1 for
three laser pulse energies. A qualitative description of the relevant physical
processes leading to cavitation and spallation was recently given in Ref.
Basko  _et al._ (2017). We summarize the crucial steps in the following,
combining it with our experimental observations. The laser pulse impact
ablates a thin (less than 1 $\mu$m) layer of tin. At the lower pulse energies
(see the 0.7 or 1.5 mJ cases presented in Fig. 1), this ablated mass is
clearly visible and seen to move away from the droplet in the direction
opposite to the laser light. The remarkably sharp outer boundary of this
ablated mass may be explained by the existence of an inhomogeneous two-phase,
gas-liquid mixture of low average density but approaching liquid density in
the vicinity of the ablation front Sokolowski-Tinten  _et al._ (1998); von der
Linde and Sokolowski-Tinten (2000). In contrast to the ns laser pulse impact
Kurilovich  _et al._ (2016), the resulting droplet propulsion is limited here.

The ablation pulse gives rise to a pressure wave starting from the laser-
impacted region of the droplet. This pressure is applied on a short, ps time
scale that is several orders of magnitude shorter than the time scale on which
pressure waves travel through the droplet.

As a consequence, the liquid is locally compressed and pressure waves travel
through the droplet Reijers  _et al._ (2017). Converging pressure ”shock”
waves superimpose in the center of the droplet where they cause tensile
stress. Once this tensile stress is above the yield strength of the liquid it
ruptures and a thin shell is formed. Theoretical modeling Basko  _et al._
(2017) has shown that there is a high sensitivity of both the rate of
expansion and the morphology of the deformed droplet to the details of the,
poorly known, metastable equation-of-state (EOS) of tin in the region of the
liquid-vapor phase transition and to the parameters of the critical point.
Thus, precise theory predictions of the post-cavitation dynamics of liquid tin
microdroplets cannot yet be made and, instead, measurements as presented in
this work may provide a sensitive instrument for probing the EOS of liquid
metals Basko  _et al._ (2017).

After passing through the droplet center, the shock wave reaches the back side
of the droplet (i.e. the side facing away from the laser) and reflects from
this free interface where it can again rupture the liquid and give rise to
spallation that rapidly propels, at several $\,100\,$m/s, a small mass
fraction of the droplet along the laser beam propagation direction (see Fig.
1) Basko  _et al._ (2017); Krivokorytov  _et al._ (2017). The fluid dynamic
description of this spallation is very rich and will be left for future work.
However, it is noteworthy that there is a regime where a strong spallation
occurs, but is mitigated under capillary action (compare the $t$ = 1 and 2
$\mu$s shadowgraphs for the 1.5 mJ case in Fig. 1).

We note that the higher-energy cases in Fig. 1 also hint at the existence of a
hole on the side of the droplet facing the laser, which may possibly be
attributed to ablation-thinning of the droplet shell. In combination with the
spallation, a ”tunnel” is thus created (see the front view shadowgraphs in
Fig. 1), that may later form a doughnut-type mass distribution. Another
scenario, when fragmentation following the collapse of the shell results in
high-speed jetting, corresponding to the lowest laser pulse energies, is
reported on in Refs. Thoroddsen  _et al._ (2009); Gonzalez-Avila  _et al._
(2011); Krivokorytov  _et al._ (2018).

In the following, we focus on the late-time dynamics set in motion by the
central cavitation. Firstly, a basic model for the time evolution of the
intact liquid shell is presented. Secondly, we study the time scale at which
first holes in this shell become visible, after which full fragmentation of
the shell rapidly sets in. This results, eventually, in a late-time mass
distribution, the understanding of which is of particular relevance for
producing EUV radiation at high conversion efficiencies.

###  III.1 Time evolution of the shell

The shadowgraphs such as those presented in Fig. 1 clearly show a
cylindrically symmetric expanding shell, the radius $R(t)$ of which is tracked
for each time step by 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 $R(t)$ data points after the laser impact, we obtain
values for the initial droplet expansion velocity $\dot{R}$($t$=0). Figure
2(a) shows a monotonic increase of this radial expansion velocity
$\dot{R}$($t$=0) as a function of laser pulse energy. Using a heuristic
argument, we collapse all obtained data onto a single curve, scaling the
obtained velocities with the initial droplet radius $R_{0}$ (see Fig. 2(b)).
To arrive at this scaling, we hypothesize that the magnitude of the induced
shock wave, and its related cavitation event, scales with the laser-facing
surface area, $\propto$ $R_{0}^{2}$ (the droplet is much smaller than the
laser spot size). Meanwhile, the expansion is impeded by the droplet mass
$\propto$ $R_{0}^{-3}$. By neglecting the possible differences in the
dissipation of the shock wave, one thus arrives at the aforementioned simple
scaling $\dot{R}$($t$=0$)$ $\propto$ $R_{0}^{-1}$. As seen from Fig. 2(b),
such a rescaling causes all data to fall on a single curve that appears to be
represented quite well by a power-law function $\dot{R}$($t$=0$)R_{0}$ = $A$
$\times$ $E^{\alpha}$ with fit parameters $A$ = 9.4$\pm 1.4$ $\rm{m^{2}}/s$
and $\alpha$ = 0.46$\pm 0.02$. Although such power-law behavior is well
established in the relevant literature on ablation Eidmann  _et al._ (1984);
Dahmani (1993); Eidmann  _et al._ (2000), these fit parameters cannot be
straightforwardly predicted from available theory, given their sensitivity to
the EOS and the details of laser-matter interaction.

Figure 2: (a) Initial radial expansion velocity $\dot{R}$($t$=0) of tin
microdroplets as a function of total laser pulse energy $E$. Yellow diamonds
are for $R_{0}$ = 15 $\mu$m and blue squares are for $R_{0}$ = 23 $\mu$m. The
error bars represent 10$\%$ uncertainty on the velocity measurements and
20$\%$ uncertainty on the energy measurements. (b) The same data, rescaled by
multiplying by the initial droplet radius $R_{0}$. The solid red line is
obtained by fitting a power-law function to the concatenated data.

Next, to describe the time evolution of the liquid droplets in Fig. 1, we
hypothesize that a small ($\sim$$\mu$m), high-pressure ($\sim$kbar) cavitation
bubble expands the liquid quickly into a thin shell. Thus, a spherically
symmetric shell expands at a certain initial velocity $\dot{R}$($t$=0), with
the cavitation pressure having done its thermodynamic work effectively at time
zero. Given that the droplet expands into the vacuum, the only limitation to
its expansion is surface tension.

The liquid tin shell has thickness $h(t)$ which decreases quadratically with
time as mass conservation implies, in the limit of a thin shell ($R(t)$ $\gg$
$h(t)$), that $h(t)=R_{0}^{3}/(3R(t)^{2})\approx
R_{0}^{3}/(3t^{2}\dot{R}(t$=0)${}^{2})$. The expansion of the shell can be
described by the Rayleigh-Plesset equation, which in the limit of a thin shell
reads Vledouts  _et al._ (2016): $p(R-h)-p(R)\approx\rho h\ddot{R}$, where $p$
is pressure. Imposing dynamic boundary conditions at the two liquid-vapor
interfaces $p(R-h)$ and $p(R)$, and noting that the only force acting on the
shell is the Laplace pressure, the net value of which is given by
$p=4\sigma/R(t)$ (in contrast to Ref. Vledouts  _et al._ (2016), where this
contribution was negligible), we find

$\ddot{R}(t)=\frac{-p}{\rho h(t)}=-\frac{4\sigma}{\rho
h(t)R(t)}=-\frac{12\sigma R(t)}{\rho R_{0}^{3}}.$ (1)

From the above expression, using the boundary conditions of a given
$\dot{R}(t$=0) and initial radius $R_{0}$, one obtains

$\frac{R(t)}{R_{0}}=\cos(\pi t/\tau_{c})+\sqrt{\frac{\textrm{We}}{12}}\sin(\pi
t/\tau_{c}),$ (2)

introducing the capillary time scale here as $\tau_{c}$ = $\pi\sqrt{\rho
R_{0}^{3}/12\sigma}$ and the relevant Weber number as We = $\rho
R_{0}\dot{R}$($t$=0$)^{2}/\sigma$. The above solution is identical in form to
that obtained previously for a disk-type expansion of a droplet following ns-
laser pulse impact Gelderblom  _et al._ (2016); Kurilovich  _et al._ (2016).

Next, we apply the theory developed above with zero fit parameters taking as
an input the experimental values for $\dot{R}(t$=0). The results are presented
in Fig. 3(b), where the experimentally obtained radius $R(t)$ (see Fig. 3a) is
rendered dimensionless by dividing by $R_{0}$ (and subtracting the unity value
offset), and plotted as a function of dimensionless time $t/\tau_{c}$. Given
the simplicity of our arguments, we find excellent agreement between model and
experiment. The observed late-time ”overshoot” that is apparent in the
experimental data shown, with theory predicting a more rapid retraction of the
shell, can be understood by considering the process of hole formation and
ensuing fragmentation, which strongly reduces the restoring forces.

Figure 3: (a) Radius $R(t)$ of shell driven by cavitation in
$R_{0}=[15]\,23\,\mu$m droplets for similar Weber numbers. [Open] black
squares: $\rm{We}=[304]\,368$; [open] brown circles: $\rm{We}=[705]$ 754;
[open] green triangles $\rm{We}=[1923]\,1778$. The inset (500 $\mu$m $\times$
700 $\mu$m) represents a side-view image of the $\rm{We}=754$ case at 2.5
$\mu$s. (b) Rescaled, dimensionless radius as a function of dimensionless time
$t/\tau_{c}$. The solid curves represent the model predictions (see Eq. (2))
for the 23 $\mu$m initial radius droplet; the dashed curves show the same but
for the smaller, 15 $\mu$m droplet. The solid gray line depicts the
destabilization radius $R(t_{*})$ as obtained from the fit of Eq. (3) to the
data (see Fig. 4), applied to Eq. (2). The data are grayed out beyond this
line.

###  III.2 Hole opening time

Figure 4:  Dimensionless time of first hole opening $t_{*}/\tau_{c}$ as a
function of Weber number for two droplet sizes: 15 $\mu$m (yellow diamonds)
and 23 $\mu$m initial radius (blue squares). For We $\lesssim$ 300, the shell
collapsed before holes were apparent. Error bars are two time steps towards
shorter values. The solid red line represents the best fit of a power law to
the concatenated data. The inset graph depicts the dimensionless radius of the
shell at the moment of hole opening. The solid black line shows the
destabilization radius as obtained from inserting Eq. (3) into Eq.( 2). The
dashed red line is based on Eq. (4) taking $\beta$ = 0.34 (see main text).

The expanding shell is subjected to Rayleigh-Taylor instabilities (RTI), here
as instabilities of the liquid-vapor interface driven by the radial
acceleration and deceleration of the liquid sheet Vledouts  _et al._ (2016).
Surface tension provides mode selection of the RTI Keller and Kolodner (1954);
Chandrasekhar (1961); Bremond and Villermaux (2005); Vledouts  _et al._
(2016). This phenomenon will lead to hole formation after which rapid hole
opening and merging will lead to full fragmentation of the shell. In our
system, there are two acceleration mechanisms that act along the surface
normal and that thus will contribute to RTI growth. Firstly, the liquid is
strongly accelerated by the cavitation pressure. Secondly, after the initial
fast expansion, the shell much more slowly decelerates under the influence of
surface tension. Holes will form at the shell piercing time $t_{*}$, when the
size of a growing instability is of the order of the shell thickness Vledouts
_et al._ (2016). In the available literature (e.g., see Refs. Vledouts  _et
al._ (2016); Bremond and Villermaux (2005)), quite generally a scaling
relation is found of the type

$\frac{t_{*}}{\tau_{c}}\propto\textrm{We}^{-\beta},$ (3)

resulting in a rough scaling for the corresponding destabilization radius
Vledouts  _et al._ (2016)

$\frac{R(t_{*})}{R_{0}}-1\propto\textrm{We}^{1/2-\beta}.$ (4)

Here we have, for simplicity, linearized the expansion rate to a constant
$\dot{R}$ $\propto$ We1/2 and in Eq. (3) dropped the usual weak scaling term
$(\eta_{0}/R_{0})^{\beta}$, where $\eta_{0}$ is the initial amplitude of RTI.
The positive power $\beta$ is typically smaller than unity Bremond and
Villermaux (2005); Vledouts  _et al._ (2016); Klein (2017). Obtaining a theory
value for this power is complicated because of the two competing mechanisms
driving RTI. We take instead the rather general form of Eq. (3) and let the
experiment provide the relevant value for $\beta$ and for the proportionality
constant. In the experiments, a hole becomes clearly visible in side view
shadowgraphy only when another hole simultaneously appears on the opposite
side. This naturally introduces a slight detection bias. However, it is
possible to reliably and reproducibly obtain an estimate for $t_{*}$ based on
optical inspection of the experimental data supported by the fact that for
times $t$ $>$ $t_{*}$ (where we take 100 ns time steps) the shell becomes
permeated with holes. We observe a monotonically decreasing value for $t_{*}$
with increasing Weber number as expected from Eq. (3). Furthermore, $t_{*}$ is
seen to decrease with droplet size for similar Weber number. Rescaling
$t_{*}\to t_{*}/\tau_{c}$ in Fig. 4 indeed collapses the data for the two
different droplet sizes onto a single curve. The resulting fit values of the
power law Eq. (3) are given by $\beta$ = 0.34(4) and 3.0(8) for the
proportionality constant. As we clearly find $\beta$ $<$ 1/2 we can now
immediately conclude from Eq. (4) that the destabilization radius $R(t_{*})$
increases with Weber number, and thus with laser energy. The obtained fit
values will serve as input for predicting the late-time mass distribution
produced by the laser impact.

Figure 5:  Phase diagram depicting the maximum obtainable radius as a function
of Weber number. White arrows illustrate the various spallation regimes (see
main text). The dashed line is obtained by maximizing Eq. (2). The solid line
shows the destabilization radius as obtained from inserting Eq. (3) into Eq.(
2); the area under this line is obtainable without holes occurring.

###  III.3 Late-time mass distribution

Having achieved good agreement between experiment and model, we now summarize
our findings in Fig. 5 to facilitate a more direct, industrial application of
our findings. The maximum amplitude of an intact retracting shell is given by
maximizing Eq. (2). The resulting dashed black curve in Fig. 5 is close to the
well-known scaling $\sim$$\sqrt{\textrm{We}}$ (as was pointed out in Ref.
Klein  _et al._ (2015)). The maximum obtainable shell size without any holes
(i.e. the destabilization radius) is obtained by inserting Eq. (3) into Eq.
(2) as in Fig. 3. The result is shown as a solid black line in Fig. 5. Once
this line is crossed in the expansion phase, the rapid breakup and
fragmentation of the shell, associated with large Weber numbers here,
precludes capillary collapse and the droplet target fragments ballistically
expand. For the low Weber numbers shown the shell collapses and jetting may
ensue Krivokorytov  _et al._ (2018).

In Fig. 5 we have also indicated ranges of Weber numbers that we associate
with different spallation regimes. For Weber numbers below We $\approx$ 500,
no spallation is visible at all; for 500 $\lesssim$ We $\lesssim$ 1000 the
spall is nearly fully retracted due to surface tension; and for We $\gtrsim$
1000 there is a rather abrupt and violent breakout of the spall. For We
$\gtrsim$ 1000, we find that hole formation in the ”tunnel” wall leads very
quickly to complete breakup of the shell, where the fragments continue on
their ballistic trajectories with radial velocities close to the initial
$\dot{R}(t$=0).

From an application perspective, it is interesting to study the ”final” late-
time fragment mass distribution that might serve as a target for a main laser
pulse in an actual industrial EUV-source based on tin plasma. In the following
analysis, we present some brief guidelines based on the here presented data
but keep the full analysis of the relevant fragmentation process (including
fragment size distributions) for a later, dedicated work. We find that the
final mass distribution changes very dramatically over a relatively small
range of Weber numbers as can be seen from the shadowgraphs presented in Fig.
6(a) as well as from their angularly averaged, radial projections shown in
Fig. 6(b). These projections can loosely be interpreted as a column-density
mass distribution along the line-of-sight of the backlight illumination. The
projections are not corrected for the small parallax angle or for the limited
depth of focus. Still, we expect to track a large fraction of the total
droplet mass. Similar mass distributions are found for 15 and 23 $\mu$m
droplets at comparable Weber numbers (see Fig. 6).

Figure 6:  (a) Late-time front (30∘) view shadowgraphs of expanded and
deformed droplets of different droplet sizes representing cases of similar
Weber numbers. The rectangular scale grid in the top-left corner visualizes
the effect of the $30^{\circ}$ parallax angle. (b) Inverted, angularly
averaged, radial projections of the obtained gray scale of corresponding
images from (a), which gives a qualitative measure of the radial column-
density-mass distribution. The dashed red line corresponds to the data from
$R_{0}$ = 15 $\mu$m droplets; the solid black line is for $R_{0}$ = 23 $\mu$m.

For We = 544 we observe that the mass distribution has a maximum in the center
of the image, as is to be expected from a collapse. For just very slightly
larger Weber numbers, we find instead a toroidal profile that becomes more and
more pronounced for larger Weber numbers. We attribute this observation to the
RTI-driven breakup of the ”tunnel” wall (thus preventing collapse) as well as
by a pronounced spallation that modifies the mass distribution, as was also
noted in Ref. Basko  _et al._ (2017).

The here established and quantified sensitivity of the final state mass
distribution to the Weber number can readily find application. As an example,
we note that a typical target size used in the industry is in the order of a
few hundred micrometers Fomenkov  _et al._ (2017), several $\mu$s after
prepulse impact. In our experiments, a relevant $\sim$450 $\mu$m diameter
target is, for example, obtained from a $R_{0}$ = 15 $\mu$m initial droplet
size after expanding for 2.5 $\mu$s, see Fig. 6(a) for We = 1923. In this
case, the main pulse would not find a significant mass fraction in its focus.
Thus, it may well be opportune to choose a lower Weber number, by tuning down
the laser energy, and obtain a more homogeneous mass distribution which could
improve the conversion efficiency of drive laser light into useful EUV
radiation by, e.g., improving the absorption of drive laser light.

##  IV Conclusions

We present an analysis of the cavitation-driven expansion dynamics of liquid
tin microdroplets that is set in motion by the ablative impact of a 15-ps
laser pulse. High-resolution stroboscopic shadowgraphy of the expanding tin
shells is combined with an intuitive fluid dynamic model that includes the
time and size at which the onset of fragmentation becomes apparent. This model
will aid follow-up studies of the fragmentation pathways. Good agreement
between model and experimental data is found for two different droplet sizes
over a wide range of laser pulse energies. The dependence of the initial
expansion velocity of the liquid shell on these experimental parameters is
heuristically captured, for application purposes, in a single power law. A
summary phase diagram of the expansion dynamics is presented. It covers
regimes with and without spallation, as well as a transition regime where the
spalled material is retracted under surface tension. This transition regime
enables finding conditions for a maximum shell expansion velocity with
strongly suppressed forward-moving debris. This phase diagram facilitates a
more direct, industrial application of our findings.

Further, the experimentally obtained late-time mass distributions are shown to
be governed by a single parameter, the Weber number. These studies are of
particular relevance for plasma sources of extreme ultraviolet light for
nanolithography. In such plasma sources, the tin mass distributions obtained
upon cavitation-driven shell fragmentation, as studied in this work, are shown
to be promising targets for efficient laser coupling Fomenkov  _et al._
(2017); Kawasuji  _et al._ (2017).

## Acknowledgments

We thank A. L. Klein and M. M. Basko for insightful discussions. This work has
been carried out at the Advanced Research Center for Nanolithography (ARCNL),
a public-private partnership between the University of Amsterdam, the Vrije
Universiteit Amsterdam, the Netherlands Organization for Scientific Research
(NWO) and the semiconductor equipment manufacturer ASML.

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