Water is ubiquitous in the universe. In the outer solar system, many satellites roughly the same size as or larger than moons such as Europa or Titan II are almost covered by water or ice.^[1,2] Deep in the universe outside the solar system, characteristic spectral lines emitted by water molecules were observed in nebulae formed by large numbers of stars such as the Orion nebulae M42.^[3] As a current hotspot, typical “ice giants” such as Neptune and Uranus are believed to contain significant amounts of water.^[4] Materials on these planets are under high pressure by virtue of their own gravitational attraction.

Thus, the equation of state (EOS) of water in extreme conditions is important in understanding the composition and evolution of planets and their satellites as well as their distributions of density, pressure, and temperature. So far, ANEOS^[5] and Sesame^[6] models for water have been employed to model ice giants. However, quantum molecular dynamics (QMD) calculations and density-functional-theory-based molecular dynamics (DFT-MD) simulations of water^[7–9] propose an EOS differing from ANEOS and Sesame. In fact, they were in good agreement with reported shock experiments.^[10–12] MD models, which suggest a superionic H₂O with hydrogen ions moving within a solid lattice of oxygen in the deep interior of icy planets — for instance, Uranus and Neptune — at high densities and low temperature, propose dynamo models to explain magnetic field structure and predict that the ice giants contain no ice but dissociated water at high ionic conductivity.^[10] However, studies on melting temperature of H₂O suggest that water remains in the liquid state even in deep interior conditions rather than in an ionic solid system.^[13] These discrepancies from experiments and theoretical models have great impact on explaining planetary magnetic field formation.^[14,15]

In fact, comparisons between shock experiments and theoretical EOS models on planetary interior structure rarely involve temperature, which has historically proven difficult to obtain. However, temperature is fundamental to thermodynamics and an important constraint to EOS models. Disagreement in various models’ temperatures might lead to completely different results. Experimental measurement of water’s shock temperature is much more challenging than dynamic parameters such as pressure because directly measuring it is difficult and requires absolute measurement of self-emission intensity. Kanani K et al. attempted to measure the shock temperature of precompressed water, but came into great uncertainty of about 35% at pressure up to 250 GPa.^[16] Melting temperature measurements of pressure below 100 GPa have been much reported.^[13,17,18] Recently Kimura et al. measured precompressed water’s P–ρ–T data up to 260 GPa.^[11] Millot et al. measured temperature below 1000 K at pressure up to 300 GPa.^[12] To validate EOS models, much more data and much higher temperature measurements at extremely high pressure are necessary.

Coupling static and dynamic compressions, shock experiments on precompressed samples can access states unreachable by either method alone, covering a broad range of P–ρ–T space and approaching conditions close to the isentrope of planets.^[16,19,20]

Here, we present shock temperature and optical measurements on precompressed water via laser-driven shock waves that can generate high pressure and temperature conditions comparable with ice giant planets’ interior conditions. Achieved temperatures lie between the principal Hugoniot and the principal isentrope of water via the precompression cell, thereby increasing access to icy giant planets’ interior states. Additionally, specific heat and electrical conductivity inferred from Hugoniot data and reflectivity were analyzed for understanding icy giants’ magnetic fields and layers.

Diagnostics provided measurements of shock velocity, reflectivity, and self-emission from which temperature can be extracted. Figure 3 shows diagnostic images of the one-shot experiment. Shock velocity, reflectivity, and self-emission of both materials can be achieved from VISAR and SOP systems. Time zero (t = 0) represents shock breakout time from the interface of quartz and water. In our experiments, velocities were measured to ∼ 1% precision.^[20,24,27] Reflectivity’s intensity was obtained from the VISAR image’s mean intensity near the target’s central position. The uncertainty was ∼ 10%, caused by planarity, measurements of intensity, reflectivity of aluminum surface, and system error. Concretely, the reflectivity can be expressed as R(442 nm) = R(660 nm) –d = R_base × C_T/C_base – d, where d is the difference of reflectivity between the VISAR probe beam wavelength (660 nm) and the SOP wavelength (442 nm) and can be regarded as a constant, C_T is the camera output in the experiment, R_base and C_base are the reflectivity and camera output of the aluminum base, respectively. The uncertainty of C_T and C_base is ∼ 5% caused by the streaked camera while the uncertainty of R_base is ∼ 6% calculated by the target team. The root mean square of shock breakout time was ∼ 6 ps,^[41] indicating that planarity and stability of the shock wave was fairly good and had little effect on error. Therefore, the uncertainty of reflectivity is ∼ 10% according to the error transfer formula. High reflectivity of both the shocked quartz and water that we observed suggests that they became conductive.

Figure 3(d) shows SOP’s normalized intensity in one shot. The micro-channel plate and SOP slit width were 725 V and 100 μm, respectively, so that the real intensity count of self-emission obtained by SOP was not too high or too low above background with neutral density filters to ensure the streak camera’s work at a linear range. We can see that the emission dropped dramatically as the shock front entered the water layer from the quartz. The diamond window’s spectral response and transmissivity at the spectral range of the single channel should be taken into account. Because the boundary on SOP records was not as clear as on VISAR records because of poorer temporal resolution of ∼ 100 ps, the boundary on VISAR records was adapted to determine the boundary on SOP.

Fig. 3.

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	Fig. 3. (color online) Shot 044. (a) VISAR line-image record; (b) shock velocity versus time determined by VISAR; (c) SOP image record; (d) self-emission versus time determined by SOP.

Figure 4 shows shocked water’s temperature data as a function of pressure, together with previous experimental works and EOS models. The interface’s pressure was determined by impedance matching^[28] with standard materials (aluminum or quartz), and uncertainty in pressure was ∼ 3% in our experiments.^[24,42] This technique involves abutting the sample against the standard material of known EOS and observing the shock front as it transits the interface between the standard material and the sample. A method of inversion was used in achieving the pressure at the interface.^[28] When the difference in impedance is small, the method of inversion is good in approximation, even under high pressure state. Thus, pressure up to 350 GPa was obtained as well as temperature up to 2.2 × 10⁴ K at the interface of the standard and water sample in one shot. The experimental data agreed well with the QMD curve. Because of attenuating shock, the continuous temperature versus pressure relationship can be obtained by introducing a P–U_S relation, fitting from our data and Kimura’s data^[11] in which initial pressure was ∼ 0.5 GPa and the peak pressure reached 350 GPa in this EOS model. The P(GPa)–U_S(km/s) relation of water (P₀ = 0.49∼ 0.60 GPa) can be inferred from Rankine–Hugoniot conservation equations as assuming a linear U_S–U_P relation. After fitting the data, A = −4.9136 and B = 1.00301 were obtained. This estimated relation was introduced in T–U_S relation to supplement a continuous T–P relation in our diagram. In this method, uncertainty of pressure was ∼ 5%, considering the fitting error of P–U_S. Therefore, both pressure calculation methods were applicable in the high pressure range we obtained. Estimated total uncertainty of temperature was 5%–12%, among which system error and measurement uncertainty were calibrated and calculated to be ∼ 5%. The remainder of temperature uncertainty comes from the Planck model itself.

Temperatures of precompressed water are significantly lower than those of principal Hugoniot states. Estimated continuous data and data at the shock breakout at higher final pressure (initial pressure P₀ = 0.5 GPa) are in very good agreement with our QMD-based EOS, but far from the Sesame. Continuous data and data at the shock breakout at lower final pressure (initial pressure P₀ = 0.49 GPa) are a bit higher than the model but in the range of error. These results well supported the QMD model, suggesting a superionic conductor of H₂O in icy planets’ deep interior, proposing a dynamo model to explain the magnetic field structure and predicting that ice giants contain no ice, but dissociated water at high ionic conductivity.

Fig. 4.

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Fig. 4. (color online) Our experiment’s data (pink squares) are shown with data of Lyzenga et al.[43] (dark cyan triangles) and Kimura et al.[11] (blue triangles). Also shown are theoretical models such as the Sesame model[6,11] (red dashed line for principal curve and green solid line for initial pressure at 0.5 GPa), QMD model[7–9,11] (blue dashed line for principal curve and rose solid line for P0 = 0.5 GPa).

To better understand mechanisms in high-pressure water, the isochoric specific heat C_V was extracted from our off-Hugoniot data^[44,45] shown in Fig. 5. Assuming that an infinitesimal section of the off-Hugoniot can be approximated as the sum of an isentropic compression step and an isochoric heating step, the variation of internal energy and temperature can be expressed as ΔE_H ≈ ΔE_S + ΔE_V and ΔT_H ≈ ΔT_S + ΔT_V, where subscripts H, S, and V refer to changes along the Hugoniot, isentrope, and isochore, respectively. Using definitions P = −(δE/δV)_s and Γ = −(V/T)(δT/δV)_s, where P is pressure and Γ is the Gruneisen parameter, we can then write

Fig. 5.

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	Fig. 5. (color online) The specific heat as a function of temperature.

Precompressed water’s reflectivity versus shock velocity was plotted in Fig. 6(a). Data at the interface where shock transit from the standard into the water were listed in Table 1. Estimated total uncertainty in reflectivity is about 15%–25%, mainly caused by measurements of reflectivity in aluminum, reflectivity of undisturbed materials, intensity response of the camera, and system error. Our reflectivity data are lower than Celliers’ principal Hugoniot data^[46] at low pressure range while quite agreeing with them at high pressure. Also, our data are quite agree with Kimura’s^[11] at low pressure while ∼ 40% higher than them at high pressure.

From Fig. 6(b), we found differences in reflectivity from different wavelengths. For the case in our experiments, emissivity ε used to determine temperature was calculated by ε = 1 − R(660 nm). To improve precision in temperature calculation, we used R(442 nm) = R(660 nm) − d in our data processing, where d is the difference of reflectivity between the two wavelengths and 442 nm is the SOP wavelength. Corrected temperatures have already been listed in Table 1 and plotted in previously mentioned figures. The correction would cause ∼ 4% uncertainty in temperature.

Fig. 6.

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Fig. 6. (color online) (a) Reflectivity as a function of shock velocity in this work (blue squares), together with experimental data of Celliers et al.[46] (open pink circles) and Kimura et al.[11] (open green circles). Fits to our data (blue solid line) are also shown; (b) Drude reflectivity as a function of carrier density at the VISAR wavelength of 660 nm (red solid line) and SOP wavelength of 442 nm (red dashed line). Difference in reflectivity between the two wavelengths is shown by the blue line. Carrier density corresponding to the maximum reflectivity from our data is marked with a cyan dashed line.

A Drude model^[46–49] was introduced to analyze shocked precompressed water’s metallic properties. Optical measurements of strongly shocked dielectrics indicate that the shock front is a specular reflector whose reflectivity is suitable to Fresnel analysis, R = |(n_s − n₀)/(n_s + n₀)|², where n_s is the complex refractive index behind the shock front and n₀ is the refractive index in the undisturbed water sample, given by the empirical formula, n₀ = 1.332 + 0.322(ρ₀ − 1). The complex index of refraction in a Drude model is given by , where ω = 2πc/λ is the optical frequency, τ_e is the electron relaxing time, ω_p = (n_ee²/mε₀)^1/2 is the plasma frequency related to the conducting state near the Fermi surface, n_e is the effective carrier density, e is the electronic charge, m = m_e/2 is the reduced mass, and ε₀ is the permittivity of free space. Chemical potential is placed midway within the gap, and the mass of holes and electrons is assumed equal. Parameters ω_p and τ_e characterize the metallic state. Electron relaxation times in fluid systems undergoing insulator–metal transitions were assumed to be near the Ioffe–Regel limit^[50] given by τ_e = γR₀/υ_e, where R₀ = 2(3/4πN_i)^1/3 is the interparticle spacing, γ ≥ 1 and υ_e = {2kT[F_3/2(−E_g/2kT)/F_1/2(−E_g/2kT]}^1/2 is the electron velocity. k is the Boltzmann constant, E_g is the mobility gap energy in the electronic density of states, and F_m(η) = (2/π^1/2) ∫ x^m/[1 + exp(x − η)]dx. N_i is the total number of particles per unit volume. water’s Drude reflectivity can be derived as a function of n_e, as shown in Fig. 6(b). The Drude model’s absorption edge is defined by , where n_c is the critical density, exceeding which high reflectivity is produced. The high reflectivity we observed suggests that the fluid becomes conductive. For the maximum reflectivity in our experiments of R ∼ 39%, the carrier density is ∼ 1.1 × 10²² cm⁻³, which is approximately 3% of the total water number density n_max = 3.8 × 10²³ cm⁻³, corresponding to a density of ∼ 1.15 g/cm³, indicating a partially ionized water at this state.

Conversely, we used carrier density in Drude formalism to model reflectivity, n_e = 2(m_effkT/2πħ²)^3/2F_1/2(−E_g/2kT), where m_eff is the effective electron mass. The energy gap along the off-Hugoniot state is assumed a linear variation with respect to density and temperature E_g(eV) = E₀ − a(ρ/ρ₀ − 1) − b(T/T₀ − 1), where E₀ ≤ 6.5.^[51] After fitting the data, a = 2.57, b ∼ 0, γ = 1.7, and E₀ = 5.0024 were obtained. Predicted reflectivity is shown in Fig. 6(a). At this time, the maximum carrier density was approximately 1.2 × 10²² cm⁻³, which is quite consistent with the value estimated from our measured reflectivity mentioned previously.

Fig. 7.

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	Fig. 7. (color online) Conductivity as a function of temperature calculated in this work (blue solid line). Also shown are the data of Chau et al.[52] rewritten by Celliers et al.[46] (open green circles) and the Neptune isentrope line (rose solid line).

Conductivity can be estimated by applying the Drude model σ_e(ω) = (n_ee²γτ_e/m)(1 − iωτ_e)⁻¹, and the inferred electronic conductivity of water sample reaches ∼ 2000 (Ω·cm)⁻¹, as shown in Fig. 7. Also shown are the data of Chau et al.^[52] whose temperatures were rewritten by Celliers et al.^[46] since Celliers estimated the electronic contribution to the total DC conductivity and suggested that electronic conductivity would begin to dominate above 5000 K, and this has implications for conductivity in icy giants’ interiors. In comparison with the computed σ_e along the Neptune isentrope, our estimated conductivity seems to agree with the planetary isentrope and extend the curve at higher temperature up to 2 × 10⁴ K. The rise in water’s reflectivity and its metallic-like conductivity indicate that such molecular dissociation is associated with the conducting transition. This may be an important support for planetary models using conductivity data to explain generation and development of the magnetic field in the planet’s interior.

A set of experiments on measurements of shock temperature and reflectivity were carried out on water sample Up to 350 GPa and 2.1 × 10⁴ K via VISAR and SOP using a combination of static precompression and laser-driven shock compression. In order to enhance the reliability of the SOP system, two kinds of standard were applied in calibration to measure the shock temperature. Both results were in good consistency with each other. Also, the difference in reflectivity between the VISAR probe beam wavelength and the SOP channel wavelength has been taken into account in temperature calculations to improve the precision of the data. The temperature–pressure data of precompressed water are in very good agreement with our QMD based EOS that suggesting a superionic conductor of H₂O in the deep interior of the icy planets. A degressive slope gradually approaching Dulong–Petit limit at high temperature was found in specific heat capacity and high reflectivity as well as conductivity were observed simultaneously. By synthetically analyzing the temperature–pressure relation, reflectivity, conductivity and specific heat at conditions reached in this work simulating the interior of planets, we can draw a conclusion that as the pressure rises, a change in ionization appears and is supposed to be attributed to the energetics of bond-breaking in the H₂O as the material transforms from a bonded molecular fluid to an ionic state. Such molecular dissociation in H₂O is associated with the conducting transition because the dissociated hydrogen atoms contributed to electrical properties. However, the boundary location of ionic, superionic, plasma, ice or fluid is still controversial and disputable. Further studies on experiments or theoretical works are required for detailed modeling on interior structures of planets.