Water has been extensively studied because of its ubiquity and importance for a variety of fundamental processes and technologies. In its liquid phase, water exhibits many thermodynamic anomalies that originate from its well-connected hydrogen bond network.1–3 Water not only plays a prominent role as a solvent in chemistry and biology,4,5 but is also used as a reactant in energy research and technology.6 Further, water is a key ingredient for many cosmic and planetary processes.7 For instance, water and water–ice mixtures are highly abundant in ice giant planets such as Uranus or Neptune.8–10 The high pressures and temperatures in the cores of these planets can create diamond from methane,11 and the polymorphs of water under such conditions might become conductors capable of generating magnetic fields.12,13

Yet, our understanding of the fundamental properties of water remains limited. One key challenge is the accurate determination of its electrical conductivity in the so-called warm dense matter (WDM) regime that typically spans electron-volt temperatures at near-solid densities.14–16 On the one hand, this is driven by its practical importance in understanding planetary processes such as the generation of the magnetic fields of Uranus and Neptune.12,13,17 A representative example falling into this category is the recent measurement of the optical reflection of shock-compressed water.18–20 Characteristic changes in reflectivity and electrical conductivity indicated the existence of superionic water ice under planetary interior conditions.21

On the other hand, water under extreme conditions22 is also important for a vast number of x-ray free-electron-laser (XFEL) experiments in physics,23 chemistry,24 and biology,25 where the interaction of short and intense XFEL pulses will lead to warm dense conditions. Further, the transport properties of warm dense water play a crucial role in evaluating sample environments and their effect on the dynamics under study. In particular, the electrical conductivity is determined by the mobile-electron density, the ionic structure, and the interaction between electrons and ions.26 An intense XFEL can excite an enormous number of electrons, modifying the potential landscapes of the electrons and ions. While there has been some experimental effort to understand the structural properties of water using x-ray diffraction techniques,27 measurements of its electrical conductivity under well-characterized WDM conditions are still lacking.

To determine the electrical conductivity of WDM generally requires measurements of the optical reflection and transmission of an excited sample.28–30 The AC electrical conductivity at the frequency of an optical probe pulse is deduced accordingly. These measurements require the sample surface to be optically flat. The recent development of gas-dynamic liquid nozzles allows the delivery of these liquid samples with submicrometer thickness.31 It opens a unique avenue to study the electrical conductivity of XFEL-excited WDM at high repetition rate. Here, we achieve a warm dense state of water by irradiating an ultrathin water sheet jet with λ = 13.6 nm radiation produced by an FEL at 10 Hz, reaching electron temperatures in the range of 10 000–20 000 K at the mass density of ambient liquid water.

The AC electrical conductivity was determined from simultaneous measurement of the optical reflection and transmission of ultrafast probe pulses at two different wavelengths, namely, 750 and 850 nm. Since the photon energies of these two probe wavelengths are below the bandgap between the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO) of water, as well as below the vertical binding energy of solvated electrons in water,32 we expect the measured conductivity to be dominated by the contributions from the electrons in the conduction band of the excited water.

At the highest excitation energy densities, we observed that the index of refraction falls to n = 0.7, and the free-carrier density exceeds n_e = 10²⁷ m⁻³, which is much higher than the density estimated by direct photoionization and is believed to be the result of carrier multiplication during thermalization of excited electrons. Further, the heated samples reach an optical conductivity of up to 2 × 10⁴ S/m, within two orders of magnitude of the values in liquid phase simple metals.33 In parallel, the samples become reflective and exhibit shielding of the laser light. At moderate temperatures, the experimental results agree well with theoretical calculations using density-functional theory coupled with molecular dynamics (DFT-MD).13 At temperatures exceeding 15 000 K, the measurements agree better with the predictions from Ziman theory,34 which is a manifestation of the liquid-metal-like electrical properties.

The experiment was carried out in the BL3 end-station of the Free-Electron Laser in Hamburg (FLASH) facility at the Deutsches Elektronen-Synchrotron (DESY). We utilized 225 eV extreme ultraviolet (XUV) FEL pulses to excite and heat free-standing water sheet-jet samples to a maximum energy density of 12 MJ/kg. The specular reflection and transmission of the excited water were measured simultaneously as functions of the pump–probe delay time for two individual optical wavelengths to determine the complex reflective index and the optical conductivity of the excited water. Detailed descriptions of the experiment are given below.

The free-standing ultrathin water samples were delivered by a microfluidic gas-dynamic nozzle; the details of the system are described elsewhere.31 In brief, ultrapure liquid water (HPLC Plus grade, Sigma-Aldrich) was accelerated and flattened by pressurized helium gas, forming an optically flat water leaf 300 µm in height and 80 µm in width right below the nozzle exit. The thickness of the water leaf varied from ∼1 µm at the nozzle exit position to less than 100 nm at the other end of the leaf. The sample thickness was determined as a function of position from thin-film interferometry on the reflection images. Instead of using a white light source to form color fringes,31 the constructive and destructive fringes from the probe pulses were employed for in situ measurements. The interference pattern was calculated as a function of thickness according to the approach described in Ref. 35. Because the film thickness increased continuously from the bottom to the top of the water leaf,31 we first identified the first constructive fringe from the bottom and used it as a reference point to determine the thickness at different positions based on the local reflectivity.

Figure 1 shows a schematic of the experimental setup. The FEL pulses excited the sample at normal incidence. The FEL pulse energies were measured by online gas-monitor detectors (GMDs) and varied in the range of 10–50 µJ during the measurements. The FEL pulse duration was shorter than 50 fs full width at half maximum (FWHM).36 The FEL pulses were relayed into the sample chamber through grazing angle reflections from carbon and nickel-coated mirrors that had a combined efficiency of ∼63%. These pulses were then focused onto the sample surface by an ellipsoidal mirror with a focal length f = 2 m and a reflectivity of 84% at 225 eV.36 The focal spot was measured to have an elliptical profile with FWHM dimensions of 45 (vertical) × 24 µm² (horizontal). The FEL was operated at a repetition rate of 10 Hz, which provided sufficient time between shots for the water leaf to replenish itself. This ensured that each pump–probe measurement was carried out on a fresh water sample. The absorption of the FEL pulses was measured by an yttrium aluminum garnet (YAG) screen-based detector that was located 1 m after the sample. By monitoring the FEL-induced fluorescence intensity on a YAG screen with and without the water sample, we found that the XUV transmission ratios through 200 and 300 nm thick water were (82.2 ± 2.6) and (77.4 ± 2.7)%, respectively, which are in close agreement with the values of 83.1 and 75.9% predicted by the well-known XUV transmission model.37 Because of these high transmission ratios, the FEL energy deposition can be considered to be uniform along the direction of propagation inside the sample. This allows us to calculate the absorbed energy density ΔE from the FEL fluence via the expression ΔE = αF/d, where α is the FEL absorption ratio, F is the incident FEL fluence, and d is the sample thickness.

Time-resolved reflectivity and transmissivity measurements of the excited water were obtained using 100 fs FWHM laser light at wavelengths of 750 and 850 nm. Probe pulses were generated from an optical parametric amplifier (OPA) that was driven by 800 nm, 60 fs pulses from a Ti:sapphire laser system. Owing to the nonlinear frequency conversion in the OPA system, the pre- and post-pulses of the fundamental laser pulses were improved. These probe beams were collimated by a 1 mm diameter iris that was 2 m upstream from the sample, so that the full water leaf could be uniformly illuminated and imaged. In both cases, the probe laser was S-polarized and then incident on the sample at an angle of 22.5°. The reflection and transmission images of the water leaf were recorded on two cameras. The time-resolved measurements were scanned from −2 to 50 ps after the excitation. The step size was 0.1 ps from −2 to 5 ps, after which it was increased to 1 ps, with 100 measurements collected at each time step.

For each probe wavelength, over 10 000 data shots were collected over the entire time scan. To compensate for the shot-to-shot fluctuations of the FEL, the energy of each FEL pulse recorded by a GMD was used to group the acquired data within a range of ±2.5 µJ (e.g., 10 ± 2.5 µJ). Between 1000 and 2000 independent measurements were analyzed in each pulse energy range. The reflection (R) and transmission (T) ratio values of the heated samples were derived from the recorded images. On each image, R and T were determined by comparing the relative brightness of the heated region with several reference regions. An example of data analysis is shown in Fig. 2.

In the reflection image [Fig. 2(a)], the curved thin-film interference pattern resulting from the gradient in jet thickness was flattened using a cross-correlation method as shown in the figure. The averaged pixel count in the FEL-heated region (6 × 6 pixel² in the region of interest) was normalized to the maximum and minimum intensities of the thin-film interference pattern. The absolute reflectivity was determined asR=RpeakIRFEL−IRminIRmax−IRmin,(1)where R_peak = 0.0965 is the peak reflectivity due to thin-film interference,35IRFEL is the average intensity in counts on the detector in the FEL-heated area, and IRmax and IRmin are the maximum and minimum intensities on the detector along the interference pattern. All these values were determined on the same image so that the shot-to-shot fluctuations of the probe pulse intensity were corrected.

In the transmission image [Fig. 2(b)], the pixel counts in the FEL-heated region are normalized to areas without sample, where 100% transmission is expected. The absolute transmission is obtained asT=ITFELITvCNorm,(2)where ITFEL and ITv are the average counts in the FEL-heated area and without the sample, respectively, and C_Norm is a normalization factor to compensate for the spatial heterogeneity of the probe and is determined from the data before FEL heating by the requirement that R + T = 1 when t < −0.1 ps. We set ITv=(ITB1+ITB3)/2 to balance the vacuum transmission on both sides of the water sample.

The time-dependent reflection and transmission at different excitation energy densities measured by the 750 and 850 nm probe pulses are shown in Fig. 3. Immediately after the XUV-FEL heating, we observe a sharp increase in reflectivity and a rapid decrease in transmission. These two effects result from the generation of conduction electrons. Initially, water molecules are ionized by absorption of XUV photons. However, the inelastic mean free path of 200 eV electrons in water is about 1 nm.38 Thus, the electrons collide with neighboring water molecules, transfer their energy, and promote the electrons into the conduction band. The high density of excited electrons screens the electric field of the probe pulse, causing an increase in reflection and a decrease in transmission. At the same time, the excess electrons absorb energy, some of which is lost through collisions consistent with Ohm’s law,26 resulting in R + T < 1. At higher excitation energy densities, we observe higher reflectivity and lower transmission, consistent with the increased creation of free carriers.

The substantial increase in reflectivity indicates that the free-electron density in the heated water is comparable to the critical densities of the probe pulse wavelengths, namely, 2 × 10²⁷ and 1.5 × 10²⁷ m⁻³ for 750 and 850 nm photons, respectively. At an excitation energy density of 10 MJ/kg, we calculate that the number density of electrons created by the absorption of 225 eV photons is only 2.8 × 10²⁶ m⁻³.

Shortly after the reflection reaches its peak values, we find that it quickly drops to nearly zero in a significantly shorter time than the corresponding transmission data (>50 ps). The latter indicates that the electron–ion equilibrium time is much slower than for XUV-FEL-heated hydrogen under similar temperature conditions (2–3 ps).39,40 This is expected, because the water molecules are much heavier than hydrogen. Further, the fast disappearance of the reflection is likely due to the formation of gradients on the sample surfaces.41,42 The smearing of the vacuum–water interface suppresses the reflection,43 but the absorption of the probe pulse is not affected. In the following subsections, we focus only on early-time data (<5 ps) before the gradient effects become significant.

The reflectivity and transmissivity of an isotropic thin film are dictated by its refractive index, the sample thickness, and the polarization, incident angle, and wavelength of the probe beam. To determine the refractive index for our experimental data, we first use the transfer-matrix method29,35 to calculate the corresponding reflection and transmission in 200 and 300 nm thick films over a wide range of refractive indices. Subsequently, we find the refractive indices that produce the reflection and transmission that match best with our measurements. Examples of the measurements using 750 and 850 nm probes are shown in Figs. 4 and 5, respectively, where panels (c) and (d) of each figure show the real and imaginary parts of the reflective index (n and k) as functions of delay time that correspond to the measured data in panels (a) and (b). The time step interval is 0.1 ps before 5 ps, and it increases to 1 ps at later time delay. About 10–20 individual data points were measured at each time step. The data points and the error bars represent the mean values and the standard deviations of the data measured at each time delay. In general, we find that n decreases from 1.33 to less than 1, a value that is characteristic of a plasma state. Simultaneously, k increases from zero to some fraction of unity. This high value of k indicates strong light absorption that is consistent with R + T < 1 after the FEL heating.

The real and imaginary parts of the inferred optical conductivity σ are shown in Figs. 4(e) and 4(f), respectively, for the 750 nm probe and in Figs. 5(e) and 5(f), respectively, for the 850 nm probe. The optical conductivity is a function of the complex refractive index and is given by26σ=i[1−(n+ik)2]ϵ0ω,(3)where ϵ₀ is the permittivity of free space and ω is the oscillation frequency of the probe beam. The square of the refractive index, i.e., (n + ik)², is equivalent to the complex dielectric function. The initial value of σ_r is 0, and σ_i < 0 at room temperature corresponds to typical dielectric material. We can see that upon FEL heating, both the real and imaginary parts increase to the order of 10⁴ S/m. The trajectories of the measured reflection and transmission and the determined conductivity as functions of absorbed FEL energy density are shown in Figs. 6(a) and 6(b) for 750 nm probe pulses and in Figs. 6(c) and 6(d) for 850 nm probe pulses. Near the trajectories of the heated samples, the contour lines in reflection are nearly parallel to the vertical axis, indicating that the reflection is less sensitive to changes in σ_r than to changes in σ_i under our experimental conditions. On the other hand, the transmission data are sensitive to both σ_r and σ_i. The increase in σ_r indicates the rise in conduction electron density that enhances both the plasma shielding and reflection, but the higher σ_r also enhances the absorption of electromagnetic waves within the material44 and attenuates the reflection. The competition between these effects therefore makes the reflection of the probe pulses less sensitive to σ_r.

To gain more insight into our experimental results, we used two different theoretical methods to calculate the electrical conductivity of the XUV-FEL-heated warm dense water. The first method was an implementation of the Kubo–Greenwood formalism based on density-functional theory coupled with molecular-dynamics simulations (DFT-MD).45 The other method made use of the Ziman theory34 of electrical conductivity based on an estimate of the electron density from DFT-MD and the total ionic structure factor computed from classical MD simulations.46

We performed DFT-MD simulations for water at the 1 g/cm³ isochore with the Vienna Ab initio Simulation Package (VASP)47–50 using 54 molecules and the Perdew–Burke–Ernzerhof (PBE) functional,15,51 similar to what was done in earlier work.13,52 To describe the ultrafast FEL experiment, we ran two-temperature simulations with a constant ion temperature of T_i = 300 K and varied the electron temperature between 300 and 40 000 K. From the DFT-MD simulations, we obtained the isochoric heat capacity C_e of the electrons by numerical differentiation of the internal energy as shown in Fig. 7(a). The relation between the peak electron temperature Tepeak and the absorbed FEL energy density ΔE isΔE=∫300 KTepeakCe(Te)dTe.(4)It is worth noting that we assume all of the absorbed energy remains in the electron system within the first 0.5 ps after FEL heating. This assumption is supported by the long time scales of the measured reflection and transmission, which suggests that the electron–ion energy relaxation time scale is of the order of tens of picoseconds. Figure 7(b) shows the result for ΔE vs T_e.

The frequency-dependent conductivity σ_r(ω)45,53 was also calculated via the Kubo–Greenwood formalism54,55 using the ionic configurations of all DFT-MD simulations and the Heyd–Scuseria–Ernzerhof (HSE) functional.56 The imaginary parts of the conductivity at the probe laser wavelengths of 750 and 850 nm can also be determined using the Kramers–Kronig relations, analogously to the procedure described in Refs.57,58, but it may be less accurate compared with the real part in the WDM regime.29,59

The real part of the optical conductivity is the sum of the free-carrier contribution and the contributions from interband transitions.28 In the case of water, the photon energies of both 750 and 850 nm photons are below the interband transition energies and are insufficient to ionize conduction electrons to the continuum.32 Thus, the Drude model can be employed to describe the free-carrier conductivity:26σr=nee2τme[1+(ωτ)2],(5)where n_e is the carrier electron density, e is the unit charge, m_e is the electron rest mass, and τ is the electron relaxation time. The electron density as a function of temperature can be determined from the occupied electron density of states (DOS). We calculated the electron DOS using DFT-MD at various electron temperatures. Results for T_e at 300 and 20 000 K when T_i = 300 K are shown in Figs. 8(a) and 8(b), respectively. The values of n_e at different electron temperatures T_e can be determined by the integral of the Fermi distribution function fTe(E) times the electron DOS gTe(E):n(Te)=∫μEmaxfTe(E)gTe(E)dE,(6)where μ is the chemical potential, which is determined by conservation of the total number of valence and conduction electrons.59 The T_e-dependent carrier density is shown in Fig. 9(a). At T_e = 5000 K, n_e is negligible. It starts to increase significantly at higher temperatures, reaching 5 × 10²⁷ m⁻³ at T_e = 20 000 K, which corresponds to an absorbed energy density of 11.6 MJ/kg. This electron carrier density is equal to 2.5 times the critical density for 750 nm light, confirming that our observed increase in probe beam reflectivity is a result of a high density of conduction electrons. Figure 9(b) shows the electron degeneracy parameter Θ, which is the ratio of the thermal energy to the Fermi energy. Because n_e increases rapidly as a function of T_e, the excited electrons become more degenerate at higher temperatures.

The electron relaxation time τ is the inverse of the scattering frequency υ. We note that υ is very sensitive to changes in temperature and density. For our experimental conditions, the contribution from electron–electron scattering is small, because the conduction electron density is still low compared with the atomic density (10²⁹ m⁻³). Thus, υ is dominated by electron–ion scattering, i.e., υ ≈ υ_ei. The Ziman theory of conductivity enables determination of υ_ei based on knowledge of the ionic structure factor S_ii according to60υei=nime2pF3∫02kF4πe2Q2+ke22Q3Sii(Q)dQ2π,(7)where Q is the momentum transfer vector, k_e is the electron screening vector, which is determined from the electron temperature and density,61n_i is the molecular density, m_e is the electron rest mass, and k_F and p_F are the Fermi wave vector and momentum of the free carriers, respectively.

To estimate the structure factor of liquid water, S_ii, classical MD simulations were performed at different temperatures under the condition of thermal equilibrium, i.e., T_e = T_i. Here, the use of classical MD allows us to model a much larger simulation box so that the lower-Q range of S_ii can be accessed, which is needed to compute υ_ei using Eq. (7). The simulation cell contained 1024 water molecules with density ρ = 0.999 g/cm³ and was performed under the constant-temperature, constant-volume (NVT) ensemble using the TIP4P/2005 force field.62 At each temperature, the system was equilibrated for 1 ns, and configurations were collected over the following 500 ps. The ion–ion structure factor was computed viaSii(Q)=1N〈∑p=1N∑q=1Ne−iQrpq〉,(8)where Q is truncated at 2π/d ≈ 0.2 Å⁻¹ with a box size of d = 31.3 Å, and r_pq is the distance between ions p and q.

The S_ii results for four different temperatures are shown in Fig. 10. Our result at room temperature is verified with data reported for the O–O structure factor measured by X-ray scattering.63 The average difference at each Q is calculated to be within 4.6%. As the temperature increases, we see that the primary liquid peak moves toward higher Q, while its peak height does not seem to change significantly. According to MD simulations, the increase in temperature stretches the correlation peaks in real space, which results in a contraction of the first atomic peak in the total radial distribution function. This is manifested by the primary liquid peak shifting to high Q, as observed in Fig. 10.

Figure 11 shows a comparison of the experimental data at 750 and 850 nm with theoretical calculations from DFT-MD and the Ziman theory. The experimental data at time delays between 0.2 and 0.6 ps are averaged. At these time delays, most of the absorbed FEL energy should be in the electron system, while the molecules remain cold. Accordingly, both calculations are carried out using the peak electron temperatures at the corresponding energy densities [see Fig. 9(a)] and the ion temperature at 300 K. The σ_r measured at both 750 and 850 nm increases as a function of T_e. The 850 nm conductivity is higher than that at 750 nm. This is a typical Drude-like behavior according to Eq. (5), i.e., σ_r decreases as the frequency of the electric field ω increases. We observe a similar trend in the theoretical calculations. The DFT-MD calculations agree well with the measurements up to T_e = 15 000 K, especially with the 850 nm data. At higher temperatures, the DFT-MD calculations overestimate σ_r. By contrast, the Ziman theory calculations underestimate the results at T_e < 15 000 K, but start to show better agreement at higher temperatures.

The failure of the Ziman theory at low temperatures is likely due to the low electron degeneracy under these conditions, i.e., Θ ≫ 1, as shown in Fig. 9(b). In this regime, there is no well-defined Fermi surface of the electrons. As the electron temperature increases, the conduction band is populated, and Θ approaches unity near T_e = 20 000 K. Consequently, the heated water behaves more like a liquid metal, and its conductivity can be better described by the Ziman theory. Note that as T_i increases, the S_ii data below Q = 2k_F also increase, as shown in Fig. 10. This will raise υ_ei, as indicated in Eq. (7). At the same time, the conduction band electron density will decrease as energy couples to the ions. These two effects together in turn reduce the magnitude of σ_r, resulting in less agreement with the experimental data, i.e., σ_r will drop by ∼20 and ∼45% if we assume T_i = 500 and 1000 K, respectively, in the Ziman theory calculations. The DFT-MD calculations properly describe the weakly degenerate conditions at low temperatures. However, at this point, it is not clear why they do not work at higher temperatures. It is possible that the exchange and correlation functional used in the DFT cannot correctly model the chemical processes under the strongly excited conditions attained in our experiment. Therefore, our measurements provide valuable data to test these theoretical calculations, especially in nonequilibrium states produced by ultrafast laser or FEL excitation, where the electron temperature is significantly higher than the ion temperature.29

Using XUV-FEL pulses to excite optically flat ultrathin water samples, we have demonstrated the first optical conductivity measurements of isochorically heated warm dense water at a continuous 10 Hz repetition rate. We have acquired the conductivity data at the first 0.5 ps following the FEL excitation, when electron temperatures reach up to 20 000 K while the ions remain cold. The results have allowed us to test nonequilibrium calculations using DFT-MD and the Ziman theory. We have found that although the DFT-MD calculations can describe the conductivity results at electron temperatures below 15 000 K, further effort is needed to improve their accuracy to predict the conductivity at higher temperatures. Instead, agreement with the Ziman theory calculations is found when T_e approaches 20 000 K, which indicates that the strongly excited water exhibits liquid-metal-like behavior when a large number of electrons are excited to the conduction band. Our results bring new insight into a WDM regime that is relevant for both high-energy-density physics and ultrafast studies in chemistry and biology. In the future investigations, we plan to acquire conductivity data with longer time delays so that the samples approach equilibrium conditions. These data could be obtained by transmission ellipsometry or interferometry measurements.

Acknowledgment. This work was supported by the U.S. Department of Energy, Office of Science, Fusion Energy Science under Grant No. FWP 100182. C.B.C. acknowledges partial support from the Natural Sciences and Engineering Research Council of Canada (NSERC). J.D.K. and D.P.D. are supported by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences under Contract No. DE-AC02-76SF00515. M.Z.M. acknowledges partial support from the U.S. Department of Energy, Laboratory Directed Research and Development (LDRD) Program at SLAC National Accelerator Laboratory, under Contract No. DE-AC02-76SF00515. M.F. and R.R. thank the Deutsche Forschungsemeinschaft (DFG) for support within the Research Unit Grant No. FOR 2440. The DFT-MD calculations were performed at the North-German Supercomputing Alliance (HLRN) facilities. The work of S.L. was supported in part by the U.S. Department of Energy, Office of Science, Office of Workforce Development for Teachers and Scientists (WDTS) under the Science Undergraduate Laboratory Internships (SULI) Program. C.R. acknowledges support from the LOEWE Excellence Initiative of the State of Hessen.