Electrical conductivity of warm dense silica from double-shock experiments

Understanding materials behaviour under extreme thermodynamic conditions is fundamental in many branches of science, including High-Energy-Density physics, fusion research, material and planetary science. Silica (SiO2) is of primary importance as a key component of rocky planets’ mantles. Dynamic compression is the most promising approach to explore molten silicates under extreme conditions. Although most experimental studies are restricted to the Hugoniot curve, a wider range of conditions must be reached to distill temperature and pressure effects. Here we present direct measurements of equation of state and two-colour reflectivity of double-shocked α-quartz on a large ensemble of thermodynamic conditions, which were until now unexplored. Combining experimental reflectivity data with numerical simulations we determine the electrical conductivity. The latter is almost constant with pressure while highly dependent on temperature, which is consistent with simulations results. Based on our findings, we conclude that dynamo processes are likely in Super-Earths’ mantles.


Supplementary Notes 1. SHOCK-MATERIAL VELOCITY RELATION ALONG THE PRINCIPAL HUGONIOT OF α-QUARTZ
In order to ensure a direct optical access to the double-shocked state, the first shock leaves the loaded sample in a transparent state. Therefore, the apparent velocity measured by the VISARs U 1 app has to be linked to U 1 p through: where n 1 is the shocked refractive index and U 0 s is the first-shock velocity [1]. Solving equation 1 for U 1 p requires the knowledge of the U s -U p relation along the principal Hugoniot of α-quartz and of the behaviour of the shocked refractive index along the same curve.
In the pressure range from 0.9 − 16 Mbar, the U s -U p relation along the principal Hugoniot α-quartz is very well expressed by a fit by Knudson & Desjarlais (2009) [2]. For pressures lower than the validity range of the fit, we built an interval-defined linear function to match the experimental data in the literature [3][4][5][6][7][8][9]. When shock-compressed up to very low material velocities (U p < 0.8 km.s −1 ), α-quartz remains in the initial phase and the U s − U p relation is linear with a positive slope. At U p < 0.6 km.s −1 , due to the effects of the material strength, α-quartz exhibits a composite wave structure, with an elastic precursor wave and a subsequent plastic compression wave; such effects have not been considered in this fit.
Consistently with a previous analysis [10], in the interval 0.8 km.s −1 < U p < 2.4 km.s −1 shock-compressed quartz exhibits a mixed phase, as a direct α-quartz → stishovite phase transition is taking place. In this region, the shock velocity displays a constant value, around 5.7 km.s −1 . For higher U p > 2.4 km.s −1 , the phase transition is complete and the phase of the shocked sample is stishovite. The slope of the U s − U p relation is positive again.
Supplementary Figure 1 shows our fit of the U s − U p relation as well as the high-pressure fit performed by [2] compared to the experimental data.

Supplementary Notes 2. MEASUREMENT OF THE REFRACTIVE INDEX OF SHOCKED α-QUARTZ
In the context of double-shock experiments, solving equation 1 to extract the real material velocity after the first shock U 1 p from the apparent velocity U 1 app measured by the VISARs requires to know not only the U s -U p relation along the Hugoniot, but also of the behaviour of the shocked refractive index as a function of the material velocity, n(U p ) [so that n 1 = n(U 1 p )]. The refractive index can also be expressed as a function of another variable, e.g. the density, provided that such variable can be determined from U p . After these relations have been applied, equation 1 becomes an implicit equation for U 1 p and can be solved.

A. Present knowledge and our work
The α-quartz ordinary refractive indices at ambient pressure (1.5469 and 1.5341 at 532 nm and 1064 nm, respectively) are well known [11]. Very precise refractive index measurements up to 60 kbar are available [12,13], showing a linear relation between refractive index and density. However, the action of the first shock in our double-shock compression method brings the sample to considerably higher pressures of a few 100 kbar, spanning the whole transparency range of the principal Hugoniot curve of α-quartz. At such conditions, no refractive index data were available at the beginning of this work.
In order to extend the range along the Hugoniot in which the refractive index of quartz is known, we performed an ancillary experimental campaign at the LULI2000 facility. We generated a uniform, transparent shocked state in an α-quartz sample by focusing a 10 ns long, weak (∼ 10 12 W/cm 2 ) laser pulse on a multi-layer target. The thermodynamic state and the actual material velocity reached in the α-quartz sample have been obtained through an impedance mismatch analysis with the aluminum layer and, for target type (b), the lithium fluoride sample. The refractive index has then been determined from the relation between the real and apparent material velocity in the α-quartz sample. At the shock arrival at the rear-side aluminum surface, the free-surface velocity of aluminum U fs (Al) and the apparent material velocity of the aluminum/α-quartz interface U app (SiO 2 ) have been measured. The material velocity in aluminum has been estimated from the measurement of U fs (Al) by approximating the adiabatic release from the shocked state to the free-surface state (P fs = 0) using the mirrored Hugoniot curve, that is, the symmetric curve with respect to the vertical line U p = U p (Al). Such approximation, employed as no adiabatic release data are available in this region, is justified by the fact that the entropy increase along the Hugoniot at low pressures is negligible. The approximation gives U p (Al) = U fs (Al)/2. From U p (Al), the other variables of the Hugoniot state reached in aluminum have been then obtained using the SESAME table 3713, which has been reproduced quite precisely by experimental studies [14]. As the shock impedance of quartz is lower than in aluminum, when the shock crosses the aluminum/quartz interface an adiabatic release takes place in the aluminum layer. The mirrored aluminum Hugoniot curve in the P -U p plane has been used to approximate such release in aluminum towards state reached in α-quartz. The α-quartz Hugoniot curve from our fit (see Section Supplementary Notes 1) has been used. The intersection between the aluminum release and the quartz Hugoniot gave the reached (P, U p ) point in shocked quartz. A scheme of the impedance mismatch analysis is provided in Supplementary Figure 3 (a). The refractive index of shocked quartz has then been extracted by inverting equation 1: For this type of target, we measure the apparent material velocity of the aluminum / LiF and of the aluminum / α-quartz interface, U app (LiF) and U app (SiO 2 ), respectively, at the shock arrival at the rear-side aluminum surface. The principal Hugoniot and shocked refractive index of LiF are known from SESAME Finally, the (U p , P ) state reached in quartz has been measured via an impedance mismatch analysis between aluminum and quartz. A scheme of the analysis is provided in Supplementary Figure 3 (b). The refractive index of shocked-quartz has been measured using equation 2.

C. Results
Our results (provided in Supplementary Table I) show that the α-quartz refractive index dependence on density can be reasonably assumed to be linear up to 4.2 g.cm −3 (320 kbar).
Supplementary Figure 4 shows the experimental results of this work together with previous low-compression data from previous studies [12,13]. The linear fits performed on a previous data set [12] and on the results of this work, both forced to reproduce the known refractive index value at initial density, exhibit surprisingly similar slopes.

Supplementary Notes 3. EXPERIMENTAL SETUP OF THE MAIN CAMPAIGNS
In our experimental campaigns, drive laser pulses at 527 nm were delivered by the North and South chains of the LULI2000 facility (École Polytechnique, France). The first pulse delivered 70 − 130 J within a duration of 10 ns; the second one 300 − 500 J in 1.5 or 2 ns.
Phase plates guaranteed a uniform illumination of the target over a 500 µm diameter.  The reflectivity of the first shock front was negligible (≤ 1%), due to the low change in the refractive index in the first compression. We could therefore assume that we essentially measured the second-shock front reflectivity R 2 (ω) at the two probe laser frequencies ω = ω L , 2ω L (corresponding to the wavelengths of 1064 nm and 532 nm, respectively). At this point, the complex optical conductivity of state 2,σ 2 (ω), cannot be univocally determined without further assumptions. Indeed, the system to be solved is composed only by one equation linking the measured reflectivity to the complex optical conductivity, but has two unknowns, [σ 2 (ω)] and [σ 2 (ω)]. Therefore, an additional relation including the unknowns must be considered. In the Drude model, such relation is implicitly provided by the functional form of the frequency dependence of the complex conductivity: where τ is the electron-ion scattering time and σ(0) is the (purely real) DC conductivity.
We wanted to find a more significative relation [σ 2 (ω)] and [σ 2 (ω)] at both probe laser frequencies (ω = ω L , 2ω L ) without the use of the simplistic Drude approach. To do so, we have first interpolated the ratio between [σ 2 (ω)] and [σ 2 (ω)] along the fused silica and stishovite Hugoniot conditions explored by the ab initio simulations from [16] and of this work, respectively (see Supplementary Figure 11), which we have previously validated by comparing our reflectivity data to their predictions. The calculated ratios are shown in Fig. 3b of the main text as a function of temperature. We performed a fit on both datasets with the function where is the opposite of the ratio between the imaginary and real part of the conductivity at a given probe laser frequency.
To estimate r(ω) to our conditions, which are off both the fused silica and the stishovite Hugoniot curves, we considered their pressure and temperature dependence. In particular, the effect of pressure on r(ω) appears to be weak. Therefore, we operated a linear interpolation between the values from [16] and this work along isothermal lines: where r m (ω, P, T ) is the estimated ratio at a certain frequency for a pressure P and a temperature T , r L (ω, T ) and r S (ω, T ) are the ratios given by the fit on the calculations from Laudernet et al. (2004) [16] and this work, respectively, and P stishovite (T ) and P fused (T ) are the Hugoniot pressure of stishovite and fused silica at the temperature T , respectively. This interpolation has physical meaning for P fused (T ) < P < P stishovite (T ), which is always the case for our data.

Results
We could write the conductivity as: where P 2 and T 2 are the measured pressure and temperature of the double-shocked state,  [16] or in this work, we extracted the ratio between the static conductivity and the real part of the conductivity at ω = ω L and ω = 2ω L : In order to determine the temperature dependency of such ratio, we fitted the datasets of .
The free parameters of the fit are s in (ω), s sat (ω), and T sc (ω). From now on, the best fits on the datasets of Laudernet et al. (2004) [16] and this work will be called s L (ω, T ) and s S (ω, T ), respectively. The datasets and the corresponding fitting functions are provided in .
At this point, the problem was to find the static conductivity that produced, via the aforementioned ratios, the closest optical conductivity values to the values obtained from the experimental data. To do so, we considered a free parameter σ m (0) representing the static conductivity and calculated the corresponding optical values σ m (ω L ) and σ m (2ω L ) for each value of that parameter via: Then, we searched for the value of σ m (0) that minimised a displacement function D[σ m (0)], defined as the sum of the squares of the displacement between the optical conductivity predicted by the model and that estimated from the experimental data, rescaled with the uncertainty associated to the latter: Since the error bars on σ(ω) are asymmetric, the upper and lower values ∆ + σ(ω) and ∆ − σ(ω) have been considered if the predicted optical conductivity value σ m (ω) was respectively higher or lower than the value estimated from the experimental data σ(ω).
In Supplementary Figure 13  Velocities directly measured by the VISARs (U 1 p and U m s ) are affected by an error due to the uncertainty on the fringe shift measure. We estimated it as 1/10 of a fringe. In the case of a material velocity measure, such as U 1 p , also the low-pressure U s − U p relation and the shocked refractive index are error sources. A Taylor expansion of the U s − U p relation thus gives the correct error estimation. The measurement of the merged shock velocity right after the merging, U m s (t 3 ), is also affected by the uncertainty associated to the linear fit on U m s (t) to extract that value. The indirect measurement of the second-shock velocity U 1 s [1] was affected by the uncertainties on the material velocity U 1 p and the timings t 1 , t 2 , and t 3 .

C. Double-shocks: thermodynamic state
Errors on density and pressure have been estimated through a Monte-Carlo routine. The analysis has been run 1000 times changing the following inputs: the thickness of the quartz sample (1 − 2 µm, indicated by the supplier), the timings t 1 , t 2 , t 3 , t 3b , and t 4 , and the velocities U 1 p and U m s (t 3b ). Errors on temperature have multiple sources: uncertainty on the SOP calibration parameter, on the measure of the counts on the SOP image, on the measure of reflectivity. Typical relative errors were of about 20%.

D. Double-shocks: reflectivity and conductivity
Uncertainties on the second-shock front reflectivity are mainly due to calibration. Other sources of error include the noise in the VISAR output and the spatial non-uniformity of the shock fronts. Typical relative error bars are of about 20%.
The estimation of the optical conductivity values is delicate and depends on several parameters. The most important error source for σ(ω) is the second-shock reflectivity measure at the same wavelength, R 2 (ω). Other error sources are the double-shocked density, and the double-shocked state pressure and temperature (which influence the estimation of the ratio between imaginary and real part of the conductivity, as detailed in Section Supplementary Notes 5 A 1). The errors on electrical conductivity have been estimated using a Monte-Carlo routine. The routine repeated the analysis 5000 times varying the following inputs: ρ 2 , T 2 , P 2 , R 2 (ω L ), R 2 (2ω L ). Supplementary Figure 15 shows a subset of the conductivity output data as a function of the reflectivity input.
The same Monte-Carlo routine employed for the determination of the uncertainties on the optical conductivity values has been used to calculate the error bars on σ(0). They are reported in Supplementary Table II. and simulations (S) on warm dense silica. ω L indicates a wavelength of 1064 nm, 2ω L a wavelength of 532 nm. The real and imaginary parts of the conductivity at each wavelength is shown together with its positive and negative error bar. Conductivity values are expressed in 10 5 S/m.    The Monte-Carlo real part of the conductivity at that wavelength is shown as a function of the Monte-Carlo reflectivity value at the same wavelength.