Laser-driven shock compression of “synthetic planetary mixtures” of water, ethanol, and ammonia

Water, methane, and ammonia are commonly considered to be the key components of the interiors of Uranus and Neptune. Modelling the planets’ internal structure, evolution, and dynamo heavily relies on the properties of the complex mixtures with uncertain exact composition in their deep interiors. Therefore, characterising icy mixtures with varying composition at planetary conditions of several hundred gigapascal and a few thousand Kelvin is crucial to improve our understanding of the ice giants. In this work, pure water, a water-ethanol mixture, and a water-ethanol-ammonia “synthetic planetary mixture” (SPM) have been compressed through laser-driven decaying shocks along their principal Hugoniot curves up to 270, 280, and 260 GPa, respectively. Measured temperatures spanned from 4000 to 25000 K, just above the coldest predicted adiabatic Uranus and Neptune profiles (3000–4000 K) but more similar to those predicted by more recent models including a thermal boundary layer (7000–14000 K). The experiments were performed at the GEKKO XII and LULI2000 laser facilities using standard optical diagnostics (Doppler velocimetry and optical pyrometry) to measure the thermodynamic state and the shock-front reflectivity at two different wavelengths. The results show that water and the mixtures undergo a similar compression path under single shock loading in agreement with Density Functional Theory Molecular Dynamics (DFT-MD) calculations using the Linear Mixing Approximation (LMA). On the contrary, their shock-front reflectivities behave differently by what concerns both the onset pressures and the saturation values, with possible impact on planetary dynamos.


S1. Target cells
Target cells are made of aluminium and measure 15 × 15 × 7.5 mm. The cylindrical liquid volume in the laser-diagnostics axis has a 2 mm diameter and is 4 mm thick. Two lateral holes, plugged with capillaries, enable the filling of the cell. Target conception has been realised at the Observatoire de Paris. The ablator/pusher combination (CH / Al / (Au) / SiO 2 ) has been glued on the drive laser side window of the cell, while a SiO 2 window has been glued on the diagnostics side. Thickness of glue was around 1µm. The plastic we used was polystyrene. The two copper or aluminium capillaries of the target cell have been glued onto the corresponding holes of the cell and connected to plastic tubes (see Figure S1). The mixture has then been injected in the cell through air suction using a pump (GEKKO experiment) or a syringe (LULI experiment). Finally, the capillaries have been cut making the cell globally watertight.

S2. VISARs
VISARs (Velocity Interferometry System for Any Reflector) can measure the velocity of a reflective shock front U s from a Doppler-shift induced fringe displacement, according to the formula: where λ 0 is the probe laser wavelength, τ is the delay induced by a transparent etalon placed on one of the two arms of the interferometer, δ is the etalon dispersion, n 0 is the pristine material refractive index, and ∆φ the phase shift on the fringe system. The velocity per fringe parameters VPF = λ 0 /2τ(1 + δ ) of the VISARs were 4.476 and 7.432 km/s (GEKKO XII) or 15.94 and 6.08 km/s (ω and 2ω VISARs at LULI2000, respectively). The use of at least two VISARs with different VPF is mandatory since, if a single VISAR is used, ∆φ can only be determined modulo 2π. A quantity 2mπ must thus be added to the measured phase shift. This quantity can be found by imposing the superposition of the two VISAR velocities. The VISARs signals were injected to streak cameras whose time range were 20 or 50 ns. The slit opening was 50 or 100 µm.

S3. SOP calibration
At GEKKO XII, the SOP has been calibrated with the use of shocked quartz as standard, using the measured shock velocity and self-emission and a previously established T -U s relation 1 . At LULI2000, the calibration has been made with the use of a standard lamp (OSRAM Wi 17/G) with known emission temperature. The streak camera has been fired 50 times for each time window from 1 ms to 10 ns and a statistical mean value of the number of counts N c has been extracted. A linear fit on the time windowdependent number of counts N c (∆t) = a ∆t + b (where b is the intrinsic background) allowed to determine the calibration factor A: where T lamp = 2610 K is the emission temperature of the lamp.

S4. Error estimation
The error on the measured fringe shift on the VISAR output has been estimated as 1/10 of a fringe. The error on U s thus depended on the number of the 2π shifts added to the measured shift to superimpose the two VISAR signals. A typical error on U s was 3%. A Monte Carlo routine has been implemented to propagate the errors through the impedance mismatching analysis (as shown in Figure S2). Error sources were the error on U Qz s and U mix s (from the VISAR measurement and from the linear fit) and the uncertainty on the quartz Hugoniot. Each analysis run used random input from a Gaussian distribution. The error on reflectivity has different sources. The main one depends on the calibration method. The fit operated on quartz measurements has a typical relative error of about 15%. Another source derives from the background estimation of the reference and the shot image. An uncertainty of the background value propagates when background is subtracted from the value extracted from the VISAR image. This introduces a typical relative error of about 2%, weakly dependent on the reflectivity value. Temperature is obtained from both SOP and VISAR measurements (of emission intensity and emissivity, respectively). The main error sources are the SOP calibration factor, the error on the number of SOP counts, and the error on VISAR-measured reflectivity. The use of the emission to determine the temperature induces the existence of a detection limit. The detection limit temperature corresponds to a number of SOP counts that is comparable to the uncertainty on this value: N c ∆N c . It can be expressed as: Typical detection limit temperatures are about 4000 K. Typical errors on the temperature are ∼ 20%, far enough from the detection limit.

S2
S5. Study of pure water Figure S3 shows an example of diagnostics output for a shot on pure water at the LULI2000 facility. The shock -particle velocity relation along the principal Hugoniot curve of pure water is shown in Figure S4. The density at ambient conditions is ρ water 0 = 0.998 g/cm 3 and the refractive index at 532 nm and 1064 nm are 1.3337 and 1.3260, respectively 2 . For U p > 7 km/s, data in the literature do not universally agree, as the SESAME table 7154 and the work by Henry et al. 3 find a lower slope than more recent work 4, 5 . Though our low-pressure datum cannot discriminate between the two slopes, our high-pressure datum clearly agrees with a linear fit on data by Knudson et al. 4 and is not compatible with the SESAME table. Figure S5 shows the adjusted pressure p/ρ 0 over the compression factor ρ/ρ 0 along the measured and DFT-MD Hugoniot curves of water and the different mixtures. The scaling of both axes with the initial densities ρ 0 is performed to account for the different compositions of the various mixtures. The initial densities of the Hugoniot curves derived from DFT-MD are 0.998 g/cm 3 for pure water, 0.7301 g/cm 3 for the water-methane-ammonia mixture, and 0.7019 g/cm 3 for the water-methane mixture at a initial temperature of 298 K. Note, that the initial DFT-MD densities and energies of the water-methane and water-methane-ammonia mixtures are chosen such that they correspond to the initial densities of WEM and SPM, respectively.

S6. Comparison between calculations and experimental data
Overall, we find remarkable agreement between the experimental data from present as well as previous work 4-7 and the present DFT-MD Hugoniot for water, whereas there are some discrepancies between experiment and simulation for the mixtures. Our simulations for the mixtures predict compression ratios similar to that of water up to an adjusted pressure of 0.8 Mbar cm 3 /g. For adjusted pressures beyond this point, the DFT-MD Hugoniot curves predict significantly higher compression factors compared to pure water. In this range, the compression ratio calculated for the water-methane mixture is found slightly higher compared to the ternary water-methane-ammonia mixture. This systematic compression behaviour is not as evident in the experimental data, but we also find the measured compression ratios for the mixtures to be higher than for water except for a few points. This difference between experiment and calculations is most likely due to the slightly different hydrogen ratios considered in the mixtures in the calculations.  Fig. 3 in the main manuscript, the compression ratios are more sensitive to this deviation. This behaviour results from the fact that the hydrogen compression ratios are significantly higher at a given pressure as illustrated in Figure S6. In this plot, we show exemplarily the water, methane, and hydrogen Hugoniot curves corresponding to the linearly-mixed water-methane Hugoniot. The initial densities are 0.9377 g/cm 3 for water, 0.4696 g/cm 3 for methane, and 0.1019 g/cm 3 for hydrogen. Since our LMA water-methane mixture is rich in hydrogen compared to the experimentally investigated water-ethanol mixture, our simulations predict systematically too high compression ratios. If the calculated Hugoniot curves were based on a less hydrogen-rich mixture, the calculated mixture points in Figure S5 would shift to lower compression ratios. However, this does not affect our conclusions on the applicability of the LMA as can be seen by comparing both water curves in Figure S6. An adjusted hydrogen content would move all solid lines slightly to the left in this plot towards the water Hugoniot with the initial experimental density of 0.998 g/cm 3 . Both red curves would exactly agree, if the LMA would work perfectly. Therefore, our calculations containing methane instead of ethanol give an upper S3 bound for estimating discrepancies between real mixtures and the LMA.

S7. Estimation of the electrical conductivity
The electrical conductivity is one of the key parameters for the understanding of planetary magnetic fields. Indeed, a planetary dynamo can be sustained if magnetic induction dominates over magnetic diffusion. This is usually expressed by the requirement that the magnetic Reynolds number R m = µ 0 σ uL 100 (where σ is the electrical conductivity of the active planetary layer component and u and L are the velocity and length scale of the fluid motion inside the layer, respectively). In gas-gun experiments, the DC electrical conductivity can be directly measured using electrodes. This approach cannot be applied to laser shock experiments. Instead, they would need a measurement of the complex refractive index of the shocked sampleñ = n + ik since, from the wave solution of the Maxwell's equations, σ (ω) = 2ε 0 n(ω)k(ω)ω. In a restricted range of pressure and temperature the absorption coefficient α(ω) = 2ωk(ω)/c and the reflectivity (where n 0 is the real part of the un-shocked refractive index and the imaginary part k 0 = 0 as is the case for initially transparent materials) can be simultaneously measured 8 at a probe laser wavelength ω. In this case, the evaluation of the conductivity is straightforward. Nevertheless, this approach is very delicate and remains restricted to few experiments and conditions. In laser shock experiments only reflectivity at one or two probe laser wavelengths (often in the green, at 532 nm, and sometimes in the near infrared, at 1064 nm) is usually measured.
In this case, a model has to be considered in order to infer the complex refractive index and thus the electrical conductivity. A common approach considers a local response regime modified to account for bound charge carriers. Within this context, the complex refractive index is expressed as whereñ b (ω) is the contribution of the bound electrons to the refractive index at the probe laser frequency andσ (ω) is the electrical conductivity at that frequency. The DC conductivity σ (0) is often 1, 8-10 estimated assuming that the studied material follows a Drude behaviour, i.e. that the frequency dependency of the conductivity can be written as where τ is the electron-ion scattering time. However, this hypothesis introduces another unknown, the relaxation time τ, which has to be estimated depending on the thermodynamic conditions. Moreover, although no DFT-MD calculations of the frequency dependence of the conductivity are available in the literature for water or planetary mixtures, studies on other materials 11,12 have shown that a Drude-like behaviour is unlikely to be followed for temperatures lower than several ten thousand of Kelvin. We made the simplistic assumption that the conductivity dependence on frequency can be neglected from the DC value to the near-infrared and visible range: where ω L is the frequency corresponding to 1064 nm (thus 2ω L corresponds to 532 nm). To model the bound electrons contribution the the refractive index n b for the explored thermodynamic conditions, we supposed that it remains real S4 (ñ b (ω) = n b (ω)) and that it can be expressed by extrapolating a Gladstone-Dale model valid for water in the visible range 13 . We also neglected the wavelength dependence of n b between 532 nm and 1064 nm. Although this model has major limitations, it can be used to compare our data with previous data obtained with the same approach. In Figure S7 the estimated DC conductivity of water and the SPM are shown as a function of temperature. The conductivity of water appears to increase with temperature and reach a plateau around 15000 K. The estimated profile of the SPM is available for temperatures that already correspond to a quasi-saturation behaviour. At 20000 K, conductivity values are ∼ 2.2 · 10 3 S/cm and ∼ 3.4 · 10 3 S/cm for water and the SPM, respectively. The estimated conductivity of the SPM is greater than that of water in the entire explored temperature range. A different behaviour was observed in multiple shock experiments on "synthetic Uranus" 14,15 . These experiments probed colder conditions where the main contribution to the conductivity is ionic, whereas in our case the dominant contribution is electronic. This conductivity estimation highlights that the use of the transport properties of water as representative of the planetary ices of Uranus and Neptune for modelling their magnetic dynamos may be too simplistic and incorrect. Figure S1. Target cell with copper capillaries connected to the plastic tubes for filling.     Figure S4. Shock vs particle velocity relation for water. Data from our two shots and the calculations performed in this work are shown together with results from previous studies [3][4][5][6] . Additionally, the relation extracted from the SESAME  Figure S5. Adjusted pressure p/ρ 0 over compression factor ρ/ρ 0 from DFT-MD data (solid triangles) in comparison to experimental data from this work and previous work 4-7, 14, 16 . Water data are presented in blue/cyan, while binary and ternary mixtures are green and red/pink, respectively.  Figure S6. Hugoniot curves corresponding to the initial conditions of the experimentally investigated water-ethanol mixture. The LMA Hugoniot for the 7:4 water-methane mixture (solid black line) was constructed using the water and methane EOS described in the text, whose Hugoniot curves are given as solid red and green lines, respectively. The dashed red line corresponds to the experimental pure water data presented in this work. The Hugoniot of hydrogen (dashed yellow line) is given to illustrate the effect of the excess of hydrogen contained in the 7:4 water-methane mixture compared to the experimentally investigated water-ethanol mixture.  Figure S7. Estimated DC electrical conductivity of water and the SPM as a function of temperature. Data in the literature for water 6,8,9,17 and of "synthetic Uranus" 15 , similar to the SPM, are shown for comparison. The reduction along the principal Hugoniot of a fit on DFT-MD calculations of the water conductivity as a function of density and temperature 18 is also shown (blue dashed line). All data are relative to the principal Hugoniot curve of the material, except when explicitly marked ("ice VII H." means that the data are along the water Hugoniot starting from ice VII at ρ 0 = 1.6 g/cm 3 ; "off H." means that data are off the principal Hugoniot since the conditions have been obtained via multiple shocks). The data show total (electronic + ionic) conductivity, unless explicitly marked ("el." means that only the electronic contribution is shown).