## 2 Materials and Methods
The operation in open atmosphere and the upgrades to the optical setup and image processing reduced the jitter of drop diameter measurements below 50 nm peak-to-peak in eight out of nine runs. This is a several-fold improvement from the previous study on XFEL-induced spallation [17], where the drop diameter data had a jitter ranging from 100 to 250 nm peak-to-peak. The noise in the drop diameter measurements prevented a point-by-point derivation of the data with respect to the time delay, as the velocity data would have been extremely noisy. We chose to fit the diameter data with an exponentially decaying function:  
\[D_{D}(t)-\mathrm{D}_{0}(t)=\mathrm{V}_{0}\left(t-\mathrm{t}_{\mathrm{shock}} \right)\exp[-(t-\mathrm{t}_{\mathrm{shock}})/\mathrm{t}_{\mathrm{decay}}], \tag{1}\]  
where \(t\) is the time elapsed from the XFEL pulse arrival, \(D_{D}(t)\) is the measured diameter of the drop, \(\mathrm{D}_{0}(t)\) is the diameter of the drop before the shock arrival including a linear correction of the experimental drifts in the drop diameter, \(\mathrm{V}_{0}\) is the initial diameter expansion velocity upon the arrival of the shock wave, \(\mathrm{t}_{\mathrm{shock}}\) is the arrival time of the shock (measured from the XFEL pulse arrival), and \(\mathrm{t}_{\mathrm{decay}}\) is the time in which the surface velocity decays from its peak value to zero. This diameter function leads to a quasi-exponential decay of the surface velocity. It has three fit parameters (\(\mathrm{V}_{0}\), \(\mathrm{t}_{\mathrm{shock}}\), \(\mathrm{t}_{\mathrm{decay}}\)) that have a clear physical significance and can be quickly estimated from the data to provide starting values for the fitting. Appendix A contains further details about the fit function and the fitting procedure. The particle velocity in the shock, \(u_{p}(t)\), is equal to half of the surface velocity, which is itself half of the droplet diameter expansion velocity. Thus, the particle velocity is one quarter of the diameter velocity and is given by:  
\[u_{p}(t)=(1/4)\mathrm{d}/\mathrm{d}t[D_{D}(t)-\mathrm{D}_{0}(t)], \tag{2}\]  
The initial pressure of the shock, \(P_{S}(\mathrm{t}_{\mathrm{shock}})\), was derived using the Rankine-Hugoniot conditions [38] and is given by:  
\[P_{S}(\mathrm{t}_{\mathrm{shock}})=\rho_{0}u_{s}(\mathrm{t}_{\mathrm{shock}}) u_{p}(\mathrm{t}_{\mathrm{shock}}), \tag{3}\]  
where \(\rho_{0}\) is the water density before the shock arrival, and \(u_{s}\) is the speed of the shock wave when it arrives at the surface. To calculate the speed of the shock wave we used the empirical relation [39]\(u_{s}(\mathrm{t}_{\mathrm{shock}})=\mathrm{c}_{0}+2u_{p}(\mathrm{t}_{\mathrm{ shock}})\), where \(\mathrm{c}_{0}\) is the speed of sound in water prior to the shock arrival. The pressure of the incoming wave after the shock discontinuity, \(P_{S}(t>t_{\text{shock}})\), was calculated by numerically integrating of temporal derivative of the pressure, using the relation:  
\[\text{d}P_{S}(t)/\text{d}t=\rho_{0}\text{C}_{\text{C}}[P_{S}(t)]\text{d}u_{p}(t) /\text{d}t, \tag{4}\]  
where C\({}_{\text{C}}[P_{S}(t)]\) is the cylindrical phase velocity [40] in the decaying pressure wave. Equation (4) models the adiabatic evolution of the liquid after the shock discontinuity [40; 41; 42] taking into account the cylindrical geometry of the shock [43]. Equation (4) and the calculation of the phase velocity are discussed further in Appendix B.