# Generation of Large Vortex-Free Superfluid Helium Nanodroplets

Anatoli Ulmer<EMAIL_ADDRESS>Institute of Optics and
Atomic Physics, Technische Universität Berlin, Hardenbergstraße 36, 10623
Berlin, Germany  Department of Physics, Universität Hamburg, Luruper Chaussee
149, 22761 Hamburg, Germany  Andrea Heilrath  Institute of Optics and Atomic
Physics, Technische Universität Berlin, Hardenbergstraße 36, 10623 Berlin,
Germany  Max-Born-Institute for Nonlinear Optics and Short Pulse Spectroscopy,
Max-Born-Straße 2A, 12489 Berlin, Germany  Björn Senfftleben  Max-Born-
Institute for Nonlinear Optics and Short Pulse Spectroscopy, Max-Born-Straße
2A, 12489 Berlin, Germany  European XFEL, Holzkoppel 4, 22869 Schenefeld,
Germany  Sean M. O. O’Connell-Lopez  Department of Chemistry, University of
Southern California, 920 Bloom Walk, Los Angeles, California 90089, USA  Björn
Kruse  Institute for Physics, University of Rostock, Albert-Einstein-Straße
23, 18059 Rostock, Germany  Lennart Seiffert  Institute for Physics,
University of Rostock, Albert-Einstein-Straße 23, 18059 Rostock, Germany
Katharina Kolatzki  Max-Born-Institute for Nonlinear Optics and Short Pulse
Spectroscopy, Max-Born-Straße 2A, 12489 Berlin, Germany  Laboratory for Solid
State Physics, Swiss Federal Institute of Technology in Zurich, John-von-
Neumann-Weg 9, 8093 Zurich, Switzerland  Bruno Langbehn  Institute of Optics
and Atomic Physics, Technische Universität Berlin, Hardenbergstraße 36, 10623
Berlin, Germany  Andreas Hoffmann  Max-Born-Institute for Nonlinear Optics and
Short Pulse Spectroscopy, Max-Born-Straße 2A, 12489 Berlin, Germany  Thomas M.
Baumann  European XFEL, Holzkoppel 4, 22869 Schenefeld, Germany  Rebecca Boll
European XFEL, Holzkoppel 4, 22869 Schenefeld, Germany  Adam S. Chatterley
Department of Chemistry, Aarhus University, Langelandsgade 140, 8000 Aarhus C,
Denmark  Alberto De Fanis  European XFEL, Holzkoppel 4, 22869 Schenefeld,
Germany  Benjamin Erk  Deutsches Elektronen-Synchrotron DESY, Notkestr. 85,
22607 Hamburg, Germany  Swetha Erukala  Department of Chemistry, University of
Southern California, 920 Bloom Walk, Los Angeles, California 90089, USA
Alexandra J. Feinberg  Department of Chemistry, University of Southern
California, 920 Bloom Walk, Los Angeles, California 90089, USA  Thomas Fennel
Institute for Physics, University of Rostock, Albert-Einstein-Straße 23, 18059
Rostock, Germany  Patrik Grychtol  European XFEL, Holzkoppel 4, 22869
Schenefeld, Germany  Robert Hartmann  PNSensor GmbH, Otto-Hahn-Ring 6, 81739
Munich, Germany  Markus Ilchen  European XFEL, Holzkoppel 4, 22869 Schenefeld,
Germany  Deutsches Elektronen-Synchrotron DESY, Notkestr. 85, 22607 Hamburg,
Germany  Manuel Izquierdo  European XFEL, Holzkoppel 4, 22869 Schenefeld,
Germany  Bennet Krebs  Institute for Physics, University of Rostock, Albert-
Einstein-Straße 23, 18059 Rostock, Germany  Markus Kuster  European XFEL,
Holzkoppel 4, 22869 Schenefeld, Germany  Tommaso Mazza  European XFEL,
Holzkoppel 4, 22869 Schenefeld, Germany  Jacobo Montaño  European XFEL,
Holzkoppel 4, 22869 Schenefeld, Germany  Georg Noffz  Institute of Optics and
Atomic Physics, Technische Universität Berlin, Hardenbergstraße 36, 10623
Berlin, Germany  Daniel E. Rivas  European XFEL, Holzkoppel 4, 22869
Schenefeld, Germany  Dieter Schlosser  PNSensor GmbH, Otto-Hahn-Ring 6, 81739
Munich, Germany  Fabian Seel  Institute of Optics and Atomic Physics,
Technische Universität Berlin, Hardenbergstraße 36, 10623 Berlin, Germany
Henrik Stapelfeldt  Department of Chemistry, Aarhus University, Langelandsgade
140, 8000 Aarhus C, Denmark  Lothar Strüder  PNSensor GmbH, Otto-Hahn-Ring 6,
81739 Munich, Germany  Josef Tiggesbäumker  Institute for Physics, University
of Rostock, Albert-Einstein-Straße 23, 18059 Rostock, Germany  Hazem Yousef
European XFEL, Holzkoppel 4, 22869 Schenefeld, Germany  Michael Zabel
Institute for Physics, University of Rostock, Albert-Einstein-Straße 23, 18059
Rostock, Germany  Pawel Ziołkowski  European XFEL, Holzkoppel 4, 22869
Schenefeld, Germany  Michael Meyer  European XFEL, Holzkoppel 4, 22869
Schenefeld, Germany  Yevheniy Ovcharenko  European XFEL, Holzkoppel 4, 22869
Schenefeld, Germany  Andrey F. Vilesov  Department of Chemistry, University of
Southern California, 920 Bloom Walk, Los Angeles, California 90089, USA
Department of Physics and Astronomy, University of Southern California, 920
Bloom Walk, Los Angeles, California 90089, USA  Thomas Möller
<EMAIL_ADDRESS>Institute of Optics and Atomic Physics,
Technische Universität Berlin, Hardenbergstraße 36, 10623 Berlin, Germany
Daniela Rupp<EMAIL_ADDRESS>Max-Born-Institute for Nonlinear Optics and
Short Pulse Spectroscopy, Max-Born-Straße 2A, 12489 Berlin, Germany
Laboratory for Solid State Physics, Swiss Federal Institute of Technology in
Zurich, John-von-Neumann-Weg 9, 8093 Zurich, Switzerland  Rico Mayro P. Tanyag
<EMAIL_ADDRESS>Institute of Optics and Atomic Physics, Technische
Universität Berlin, Hardenbergstraße 36, 10623 Berlin, Germany  Department of
Chemistry, Aarhus University, Langelandsgade 140, 8000 Aarhus C, Denmark

###### Abstract

Superfluid helium nanodroplets are an ideal environment for the formation of
metastable, self-organized dopant nanostructures. However, the presence of
vortices often hinders their formation. Here, we demonstrate the generation of
vortex-free helium nanodroplets and explore the size range in which they can
be produced. From x-ray diffraction images of xenon-doped droplets, we
identify that single compact structures, assigned to vortex-free aggregation,
prevail up to $10^{8}$ atoms per droplet. This finding builds the basis for
exploring the assembly of far-from-equilibrium nanostructures at low
temperatures.

The visualization of dopant nanostructures formed inside a helium droplet
offers an opportunity to parse the factors driving their self-assembly in a
cold, superfluid environment, including the effect of intermolecular forces
among the dopant materials and their attraction to quantized vortices, if
present. While helium nanodroplets have long been used as a matrix for cooling
and preparing dopants for their spectroscopic studies [1, 2], it is only
through x-ray coherent diffractive imaging that the visualization of both the
droplets and dopant nanostructures has become possible [3, 4]. One testament
to this imaging technology is the captured in situ configurations of xenon-
traced vortex filaments in submicron-sized superfluid helium droplets [5, 6,
7, 8, 9, 10].

Vortices with quantized circulation are a manifestation of superfluidity and
play an essential role in the rotational dynamics of both Bose–Einstein
condensates and superfluid helium [11, 12, 13, 14]. Although these vortices
are fascinating and intriguing by themselves, their presence dominates
nanostructure formation since many dopants are easily attracted to them [15,
16, 17, 18, 9, 10]. On the other hand, in their absence, the dopants may be
randomly distributed inside the confined droplet space because of the
droplet’s superfluid state and the weak interaction between helium and the
dopant [19, 1, 20, 2]. Additionally, since any heat associated with the doping
process and dopant aggregation are rapidly dissipated through the evaporation
of helium atoms, the dopants could form amorphous particles with a fractal-
like substructure influenced by intermolecular forces [19, 21, 22, 23, 24]. In
small helium nanodroplets containing a few tens of thousands of atoms,
examples of spectroscopically-identified nanostructures include: silver
forming compact clusters at one or several sites in the droplet [20], hydrogen
cyanide assembling into linear chains [25], water forming the smallest
observed ice nanostructure with only six molecules [21], weakly-bound
magnesium aggregating in a foam structure [26, 27], and a core-shell structure
of a multi-component doped droplet [28, 29, 30].

To explore and image self-organized structures, especially those that can only
be formed in superfluid droplets, the presence of vortices needs to be
suppressed. One means of controlling their presence is to generate the
droplets by expanding cold helium gas. Here, nanodroplets may be
stochastically formed from the initial condensation of helium into small
clusters that further grow through collision with other helium clusters at
some distance away from the nozzle [31]. This way of droplet generation may
preclude the acquisition of angular momentum from shear forces prompted by the
co-flowing helium fluid. In this paper, we demonstrate that droplets produced
from gas condensation using a conical nozzle are larger and have smaller
rotational distortion as compared to those produced using a pinhole nozzle in
previous experiments [5, 32, 8, 33]. Smaller shape deformations indicate a
smaller angular momentum of the droplets, which likely contain few if any
vortices. Because of this effect, we could identify two major types of xenon
nanostructures, filaments and compact, that are respectively assigned to
vortex-induced [34, 35, 36, 6, 7, 8, 9, 10], and vortex-free aggregation.
Finally, we map the occurrence of these two structures based on the droplet
size and find that droplets smaller than $\sim 200\text{\,}\mathrm{nm}$ in
diameter are conducive to imaging far-from-equilibrium nanostructures; opening
routes for studying self-assembly in a self-bound superfluid droplet.

X-ray coherent diffractive imaging was performed at the Nano-sized Quantum
Systems end station of the European XFEL’s Small Quantum Systems scientific
instrument [37, 38]. This imaging technique takes snapshots of the droplet
size and shape and, if doped, the structure of the dopant aggregates [3, 4].
The helium droplets were produced using a conical nozzle with a throat
diameter of $150\text{\,}\mathrm{\SIUnitSymbolMicro m}$, a half-opening angle
of $None$, and a channel length of $9.3\text{\,}\mathrm{mm}$. The nozzle was
attached to a Parker valve and operated at a constant stagnation pressure of
$20\text{\,}\mathrm{b}\mathrm{a}\mathrm{r}$, while the nozzle temperature
$T_{0}$ was varied from $5\text{\,}\mathrm{K}$ to $14\text{\,}\mathrm{K}$.
Variable amounts of xenon were introduced in a gas doping cell installed along
the droplet’s flight path. At $\sim 1.05\text{\,}\text{meters}$ from the
nozzle exit, the droplets reached the interaction volume where they
intersected the x-ray beam that had a photon energy of
$1\text{\,}\mathrm{keV}$, pulse energies in the range of
$3\text{\,}\mathrm{mJ}$ to $5\text{\,}\mathrm{mJ}$, and a focus diameter of
$\sim 1.5\text{\,}\mathrm{\SIUnitSymbolMicro m}$ (FWHM) [38]. The scattered
light from a diffraction event was collected up to $None$ by an area detector
located $\sim 370\text{\,}\mathrm{mm}$ from the interaction volume [37]. A
detailed description of the experiment and other relevant information are
given in the Supplementary Material (SM) [39].

Figures 1(a) to 1(c) show examples of diffraction images for pure (i.e.,
undoped) droplets produced at $T_{0}=$5.0\text{\,}\mathrm{K}$$,
$7.3\text{\,}\mathrm{K}$ and $10.0\text{\,}\mathrm{K}$, respectively. The size
distributions are depicted in Fig. 1(d) where the droplet size
$N_{\mathrm{He}}$ is customarily given as the number of helium atoms and is
related to the droplet’s radius $R_{\mathrm{D}}$ through
$N_{\mathrm{He}}=(R_{\mathrm{D}}/$0.222\text{\,}\mathrm{nm}$)^{3}$ [1, 2]. The
nozzle temperatures explored here encompass two droplet production regimes:
the fragmentation of liquid helium close to the nozzle throat at
$T_{0}=$5\text{\,}\mathrm{K}$$, and the condensation of helium gas after the
nozzle throat at $T_{0}\geq$10\text{\,}\mathrm{K}$$ [1, 2]. The size
distribution at $T_{0}=$5\text{\,}\mathrm{K}$$ is bimodal with a broad size
range, whereas, that for $T_{0}\geq$10\text{\,}\mathrm{K}$$ is more narrow
with an almost log-normal shape. Under similar stagnation conditions, the
droplets produced at the gas condensation regime using the conical nozzle are
about two to three orders of magnitude larger than those produced using the
$5\text{\,}\mathrm{\SIUnitSymbolMicro m}$ pinhole nozzle [1, 2], also see Fig.
S7 in SM [39].

Figure 1: Selected diffraction images of pure droplets obtained at a nozzle
temperature of (a) $5.0\text{\,}\mathrm{K}$, (b) $7.3\text{\,}\mathrm{K}$, and
(c) $10.0\text{\,}\mathrm{K}$. The size distributions are given in panel (d),
and their aspect ratio distributions in panel (e). The numbers of diffraction
images used for the size (aspect ratio) distributions are $291$ ($236$) for
$5.0\text{\,}\mathrm{K}$, $766$ ($488$) for $7.3\text{\,}\mathrm{K}$, and
$272$ ($214$) for $10.0\text{\,}\mathrm{K}$.

As with rotating viscous liquid drops [40], a rotating superfluid helium
droplet can have a series of equilibrium shapes depending on its angular
velocity and angular momentum [41, 42]. One way to describe a droplet’s shape
is through its aspect ratio $AR$, which is defined as the ratio between the
semi-major and semi-minor axes of the projected droplet density onto a two-
dimensional plane. Figure 1(e) shows the aspect ratio distribution for
$T_{0}=$5.0\text{\,}\mathrm{K}$$, $7.3\text{\,}\mathrm{K}$ and
$10.0\text{\,}\mathrm{K}$ with average values of
$(1.011\,{}^{+}_{-}\,{}^{0.012}_{0.011})$, $(1.016\pm 0.013)$, and $(1.015\pm
0.012)$, respectively. Collectively, $$96.4$\%$ ($$99.9$\%$) of the droplets
generated using the conical nozzle have an aspect ratio smaller than $1.05$
($1.10$), making them close to spherical, and the largest observed $AR$ is
$1.3$. These values are considerably smaller than those obtained from previous
investigations using different kinds of nozzles. For instance, values as high
as $2.3$ have been reported for droplets produced from a
$5\text{\,}\mathrm{\SIUnitSymbolMicro m}$ pinhole nozzle at
$20\text{\,}\mathrm{b}\mathrm{a}\mathrm{r}$ and $5\text{\,}\mathrm{K}$ with
$$\sim 65$\%$ of the droplets having $AR\leq 1.05$ [43]. Another experiment
using a flow-cryostat but with the same $5\text{\,}\mathrm{\SIUnitSymbolMicro
m}$ pinhole nozzle at $T_{0}\simeq$5\text{\,}\mathrm{K}$$ reported a mean
value $\langle AR\rangle=1.059\pm 0.005$, with $1\%$ of the droplets
exhibiting $AR\geq 1.4$ [33]. For droplets produced at
$80\text{\,}\mathrm{b}\mathrm{a}\mathrm{r}$ and $5.4\text{\,}\mathrm{K}$ using
a trumpet-shaped nozzle with a throat diameter of
$100\text{\,}\mathrm{\SIUnitSymbolMicro m}$ and a half-opening angle of
$None$, aspect ratios as large as $\sim 3$ were observed, and $92.9\%$ of the
droplets have close-to-spherical shape [44]. Further elaboration about the
droplets produced using these three nozzles is included in the SM [39].

Figure 2: Major types of xenon nanostructures inside helium droplets generated
using a conical nozzle at different nozzle temperatures $T_{0}$. Shown are
selected examples of compact structures from vortex-free droplets and
filaments from vortex-hosting ones. For each structure type, the first row
shows the measured diffraction image, whereas the second shows the numerically
reconstructed xenon-doped droplets using a modified DCDI, see the SM [39]. The
$\mathrm{nm}$-scale is different in each figure. The droplet is shown in blue
colors, while the droplet density scales with the brightness. The color scales
for the diffraction and the dopant reconstruction are shown on the upper right
portion.

The consequence of generating close-to-spherical and large superfluid
nanodroplets is reflected on the types of nanostructures observed inside them.
Figure 2 shows diffraction images and numerically reconstructed xenon clusters
in droplets generated at different nozzle temperatures. The image
reconstruction algorithm is a modified version of the Droplet Coherent
Diffractive Imaging (DCDI) [6, 39]. In each reconstruction, the droplet is
represented by dark blue to blue colors, while the xenon clusters are
represented by dark red to light green colors. Two major kinds of structures
were observed: compact (upper panel) and filaments (lower panel). The compact
structures are assigned to vortex-free aggregation, are located at some
distance away from the droplet’s center, and have roughly circular shapes with
radii ranging from $10\text{\,}\mathrm{nm}$ to $15\text{\,}\mathrm{nm}$.
Because of the high thermal conductivity of superfluid helium, these compact
structures possibly grow with an amorphous or even fractal morphology as
similarly observed in bulk superfluid helium [23, 45, 24]. Compact structures
larger than $20\text{\,}\mathrm{nm}$ are also observed (see Fig. S5 in SM
[39]), but instead of having a more circular shape, they seem to be composed
of two clusters, where each may have initially been formed at different
aggregation sites before colliding and fusing into a cluster-cluster
aggregate. However, because the heat of condensation is not enough to melt
them, the two clusters remain distinct without reforming into one big clump
[23].

Similar to previous observations [5, 6, 7, 8, 9, 10], the filaments are also
assigned to xenon clusters tracing the vortex length. In contrast, however,
the number of vortices in our experiment is rather few ($<6$), and no Bragg
peaks, whose occurrence in a diffraction attests to the presence of a vortex
lattice [5, 8], were observed. The off-centered single vortices in Fig. 2
convey that the xenon distribution along the vortex line is uneven; rather,
each is dotted by distinct nanometer-sized ($10$ to $20\text{\,}\mathrm{nm}$)
xenon clusters. Additionally, the filament is not smooth and its curvature is
not as expected for an off-centered vortex [46, 15]. Instead, it has
oscillations along its length that specify underlying dynamics of how dopants
approach a vortex [16, 17, 45, 47] and indicate the presence of helical Kelvin
waves relevant to quantum turbulence and the cascade of excitations in
superfluids [11, 48, 49]. This observation of single vortices is instrumental
to understanding vortex nucleation and decay in a superfluid helium droplet,
as similarly observed in Bose–Einstein condensates [50, 51].

Figure 3: Mapping structure formation as a function of the droplet size. The
counts (left) and fractions (right) of the identified compact and filament
xenon nanostructures for $T_{0}=$5\text{\,}\mathrm{K}$$,
$7.3\text{\,}\mathrm{K}$, and $10\text{\,}\mathrm{K}$ (from top to bottom).
Structures that could not be assigned to either categories are categorized as
inconclusive (see Fig. S5 in SM [39]) and serve as identification error for
the fraction diagrams.

Crucial to distinguishing these two nanostructures is the droplet’s angular
momentum, which may be partitioned between quantum vortices and capillary
waves. Almost all droplets in earlier imaging experiments were generated in
the liquid fragmentation regime and contained multiple vortices [5, 6, 7, 32,
52, 44, 8, 33, 9, 10]. These droplets probably acquire angular momentum
through the shear flow of the expanding helium fluid [34, 32, 33]. A smaller
nozzle orifice, or higher velocities due to higher stagnation pressures of the
expanding fluid, increases the action of shear forces that translates to a
larger vorticity of the fluid flow. Within a few nanoseconds during flight,
the droplets become superfluid via evaporative cooling and reach a temperature
of $\sim 0.4\text{\,}\mathrm{K}$ after $\sim
100\text{\,}\mathrm{\SIUnitSymbolMicro s}$ [18] with the angular momentum
conserved. If the acquired angular momentum is sufficient, one or more stable
quantized vortices are nucleated. If more than enough is acquired, multiple
vortices form a triangular lattice with the same quantum of circulation and
sense of rotation [11, 14] and bear most of the droplet’s angular momentum
[41, 8, 42]. At high angular momentum, the velocity fields from the vortices
and the capillary waves induce deformations that resemble the equilibrium
shapes of rotating classical viscous droplets [41, 8, 42], i.e., from
spherical to oblate and to triaxial prolate shapes at higher angular momenta
[5, 32, 44, 8, 33]. The angular momentum required for a single straight vortex
to reside in the droplet’s center is equal to $N_{\mathrm{He}}\hbar$ [53, 46,
15, 41, 42]. If less angular momentum is present, an off-axis curved vortex
may be created that precesses around the droplet’s center and whose angular
momentum decreases with its curvature [46, 15, 16]. On the other hand, if the
acquired angular momentum is insufficient for vortex nucleation, it may be
stored as capillary waves, and, in this case, the droplets can only rotate by
adopting prolate-shape configurations [53, 41, 42]. In the present experiment,
the close-to-spherical droplets suggest that they contain small angular
momenta, consistent with the observation of the absence or a small number of
vortices, see SM for further discussion [39]. However, the exact partition of
the small angular momenta between few vortices and capillary waves requires
further theoretical and experimental studies, especially for sub-micron to
millimeter-sized superfluid drops [54].

The finding that compact structures are mainly observed in smaller droplets
and filaments in larger ones can locate the droplet sizes where the formation
of either structure is favored. Figure 3 shows the abundance and fractions of
these xenon structures for $T_{0}=$5\text{\,}\mathrm{K}$$ (top),
$7.3\text{\,}\mathrm{K}$ (middle), and $10\text{\,}\mathrm{K}$ (bottom). The
vertical dashed lines on the fraction panel indicate that the transition
crossover, or where the onset of vortex-induced xenon aggregation is
facilitated, occurs at $N_{\mathrm{He}}\approx 10^{8}$, almost independent of
how the droplets are generated. Nevertheless, the means of droplet generation
affects the relative abundance of either structures. For instance, filaments
are dominant in the liquid fragmentation regime at
$T_{0}=$5\text{\,}\mathrm{K}$$, and conversely, compact structures in the gas
condensation regime at $T_{0}\geq$10\text{\,}\mathrm{K}$$. The observation of
filaments in the gas condensation regime is interesting since it suggests that
angular momentum was acquired during droplet growth, possibly from droplet
collision. A simple kinematic model to estimate the acquired angular momentum
shows that for the same collision velocity the angular momentum per atom
increases with droplet size, and plausibly why smaller droplets are vortex
free [39]. In other words, larger droplets require slower collision velocities
to have the same angular momentum per atom as compared to smaller droplets.
Moreover, these larger droplets also need less angular momentum per atom to
maintain a stable curved vortex [15, 39]. These estimates corroborate the
observed size-dependent vortex stability in superfluid helium droplets.
Finally, at $12\text{\,}\mathrm{K}$ and
$20\text{\,}\mathrm{b}\mathrm{a}\mathrm{r}$, only compact structures were
found, but both structures with a similar crossover were observed again when
the stagnation pressure was increased to
$60\text{\,}\mathrm{b}\mathrm{a}\mathrm{r}$, see Fig. S8 in SM [39].

In summary, we have demonstrated that large superfluid helium nanodroplets
generated using a large-diameter conical nozzle are close to spherical where
the angular momentum is likely shared between a few vortices and surface
capillary waves. Additionally, we distinguished two main types of dopant
aggregation, where their relative abundance is droplet size dependent. The
size-dependent transition from vortex-free to vortex-induced structures also
emphasizes the size dependence of the rotational state of superfluid helium
droplets. For instance, xenon-traced vortex filaments are only readily
observed in droplets with more than $10^{8}$ helium atoms. This size not only
benchmarks the onset where vortex-induced nanostructure formation starts to be
dominant, but it also indicates the droplet size amenable for the preparation
of self-organized nanostructures without the influence of quantum vortices.
This study could be a beginning for x-ray imaging of far-from-equilibrium
nanostructures from various kinds of dopant materials with different
intermolecular forces (e.g. hydrogen or metallic bonding). Without the
influence of vortices, it also becomes possible to image dynamics (e.g.
nanosplasma ignition) occurring between the dopants and the droplet (in
contrast to the observation in Ref. [55]), and how charges are distributed on
the droplet’s surface [10].

###### Acknowledgements.

We acknowledge the European XFEL in Schenefeld, Germany for the beamtime
allocation (SQS 2195) at the SQS instrument, and for the user financial
support grant. We would also like to thank the staff, especially Steffen Hauf,
for their enormous assistance. We thank the precision mechanics workshops at
the TU Berlin Physics Department and Max-Born-Institut for their technical
support. Additionally, we greatly benefited from discussions about droplet
collisions, nucleation of vortices, and rotation of prolate-shaped droplets
from Manuel Barranco, Francesco Ancilotto, and Martí Pi. A.U., R.M.P.T., D.R.,
T.Möller., J.T., and B.Krebs acknowledge funding provided by the
Bundesministerium für Bildung und Forschung (BMBF) via grant No. 05K16KT3, the
BMBF Forschungsschwerpunkt Freie-Elektronen-Laser FSP-302, Deutsche
Forschungsgemeinschaft (DFG) Mo 719/13 and Mo 719/14. A.U. and M.M.
acknowledge support by the Cluster of Excellence ”Advanced Imaging of Matter”
of the DFG - EXC 2056 - project 390715994. Additionally, M.M. acknowledges
support by the DFG - SFB-925 - project 170620586. A.Heilrath, B.S., K.K.,
A.Hoffmann, and D.R. acknowledge funding from the Leibniz-Gemeinschaft via
grant No. SAW/2017/MBI4, and K.K. and D.R. further acknowledge the Swiss
National Science Foundation under the Grant No. 200021E_193642. S.M.O.O-L.,
S.E., A.J.F., and A.F.V. were supported by the National Science Foundation
(NSF) under grant nos. CHE-1664990 and CHE-2102318, and S.M.O.O-L. and A.F.V.
were also supported by NSF grant nos. DMR-1701077 and DMR-2205081. H.S. and
R.M.P.T acknowledge support from Villum Fonden through a Villum Investigator
Grant No. 25886. T.F. acknowledges support by the DFG via SFB 1477
“light–matter interactions at interfaces” (ID: 441234705) and via the
Heisenberg program (ID: 436382461). M. Ilchen acknowledges funding from the
Volkswagen Foundation for a Peter-Paul-Ewald Fellowship. B.Kruse gratefully
acknowledges funding by the European Social Fund (ESF) and the Ministry of
Education, Science and Culture of Mecklenburg-Western Pomerania (Germany)
within the project NEISS (Neural Extraction of Information, Structure and
Symmetry in Images) under grant no ESF/14-BM-A55-0007/19. Data recorded for
the experiment at the European XFEL are available at
https://doi.org/10.22003/XFEL.EU-DATA-002195-00.

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