## 2 Materials and Methods  
_X-ray free-electron laser (XFEL) parameters_. The experiment was carried out at the SACLA XFEL [19], in the EH4c experimental station on the BL3 beamline [31]. The XFEL pulses had 9.991 \(\pm\) 0.029 keV photon energy and an average pulse energy at the XFEL source around 400-500 \(\upmu\)J. The X-ray beam was focused by a set of Kirkpatrick-Baez (KB) mirrors [32] to a full width half maximum (FWHM) size of 1.3 \(\upmu\)m horizontally and 1.235 \(\upmu\)m vertically. The experiment also used synchronized femtosecond optical pulses for imaging. The imaging pulses were generated by the SACLA laser system (a Ti:sapphire chirped pulse amplifier system with 30 fs duration and 800 nm wavelength) and were delivered to the EH4c experimental station. The light pulses were frequency doubled to 400 nm to improve the optical resolution and had an energy of 450 \(\upmu\)J at the experimental setup. _Experimental setup for measuring the shock wave characteristics_. The shock waves were generated by the focused X-ray laser pulses in spherical water microdrops with 29.9-30.5 \(\upmu\)m and 61 \(\upmu\)m diameters, and the shock pressures were calculated from the velocity of the droplets' surface during the reflection of the shock wave. Figure 1a shows the experimental geometry: The center of the droplets was aligned with the XFEL beam to better than 1 \(\upmu\)m, and multiple droplets were first exposed to XFEL pulses and then imaged optically using pulsed laser illumination, to determine how the equatorial diameter of the drops varied as a function of the time delay. Differentiation of the diameter data with respect to the time delay provided the velocity of the droplet's surface in the equatorial region, where the shock waves were reflected at an angle of incidence close to 90\({}^{\circ}\), leading to an instantaneous surface velocity equal to twice the instantaneous particle velocity in the shock wave [33]. The surface position as a function of the time delays represents the primary data recorded during the experiment, and the shock properties were calculated from it. A top-view schematic of the experimental setup, which was operated at ambient temperature and pressure, is shown in Figure 1b. After the X-ray pulses were reflected by the KB focusing mirrors, they exited the beamline through three beryllium and one Kapton thin windows, and traveled a 215 mm long path in air before being focused. The droplets were overlapped with the beam in the focal plane of the X-rays and were imaged by a high-resolution microscope with a long working distance objective (Mitutoyo M Plan Apo 50\(\times\), NA = 0.55). The imaging axis was horizontal and perpendicular to the XFEL beam. An additional low-resolution imaging system was built on the same axis as the XFEL beam, using mirrors with ~3 mm diameter holes for the XFEL beam, and was used in conjunction with the high-resolution microscope to align the droplets to the beam before each run. After passing through the droplet, the energy of the spent XFEL pulse was measured for each pulse by two detectors that collected X-rays scattered from a Kapton foil, and by a third camera-type detector that also absorbed and blocked the beam. The water droplets were generated by tapered micronozzles (MJ-AT series, MicroFab Inc., Plano, TX, USA) with 15 and 30 \(\upmu\)m diameter orifices, equipped with piezoelectric actuators which induced a controlled Rayleigh breakup of the jet into monodisperse droplets. The nozzle control signal was synchronized to the XFEL pulses, to ensure that a droplet was present and properly aligned prior to the arrival of each pulse. The droplet injection was equipped with a temperature control system using circulating liquid from a temperature-controlled bath, which fed heat exchangers for the liquid sample and for a ~1 cm diameter stream of nitrogen gas coaxial with the nozzle.