Shock-wave equation-of-state measurements in fused silica up to 1600 GPa. Rochester, NY , USA. Rochester, NY 14620, USA

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1 Shock-wave equation-of-state measurements in fused silica up to 1600 GPa C. A. McCoy, 1,2 M. C. Gregor, 1,3 D. N. Polsin, 1,3 D. E. Fratanduono, 4 P. M. Celliers, 4 T. R. Boehly, 1 and D. D. Meyerhofer 5 1 Laboratory for Laser Energetics, University of Rochester, Rochester, NY , USA 2 Department of Mechanical Engineering, University of Rochester, Rochester, NY 14620, USA 3Department of Physics and Astronomy, University of Rochester, Rochester, NY 14620, USA 4 Lawrence Livermore National Laboratory, Livermore, CA 94550, USA 5 Los Alamos National Laboratory, Los Alamos, NM 87545, USA The properties of silica are important to geophysical and high-pressure equation-of-state research. Its most-prevalent crystalline form, -quartz, has been extensively studied to TPa pressures. This article presents Hugoniot measurements on amorphous silica, commonly referred to as fused silica, over a range from 200 to 1600 GPa using laser-driven shocks and an -quartz standard. These results extend the measured Hugoniot of fused silica to higher pressures. In the 200- to 600-GPa range, the data are in very good agreement with those obtained by Qi et al. [Phys. Plasmas 22, (2015)] using magnetically driven aluminum impactors and aluminum as a standard material. A new shock velocity/particle velocity relation is derived to fit the experimental data. 1

2 I. INTRODUCTION Silica is the single most-abundant compound in the earth s mantle and crust and an end member of the MgO-FeO-SiO 2 system that forms those structures. 1 3 Its high-pressure behavior is fundamental for studies of interest to geophysics, tectonophysics, and the formation of exoplanets. 4 In both amorphous and crystalline forms, it is a prototype for studying materials at extreme conditions using both static 5 8 and dynamic 9 13 compression experiments. Many of these experiments used synchrotron radiation to probe various solid solid phase transitions and map the melt curve. 5 7,12,13 The variety of stable polymorphs that exist at ambient pressure makes it possible to cover a wide region of phase space with single-shock techniques. This made it possible to approximate the Grüneisen parameter for solid silica Until recently, most high-pressure data for silica were for pressures below 200 GPa, exploring melting and different phase changes in the structure. The development of drivers and techniques that produce higher pressures made it possible to study fluid silica produced by TPa shocks using various polymorphs of silica. Understanding material behavior at these extreme conditions is of interest for planetary astrophysics, equation-of-state (EOS) research, and inertial confinement fusion. The principal Hugoniot and release isentrope of -quartz were measured in the fluid regime (>100 GPa). Research has also been conducted in fused silica, 19,23 26 stishovite, 24 and low-density silica foams This furthered the understanding of how these materials behave at the core mantle boundary on Earth at pressure temperature conditions where silica would be fluid. 30 At higher pressures (>300 GPa), the 2

3 compressed-silica EOS describes the behavior of material at the core mantle boundary in super-earths and other giant exoplanets. 24 For further increases in pressure along the Hugoniot, silica transitions from a bonded molecular liquid to a dissociated atomic fluid. 19 Alpha quartz is frequently used as an impedance-matching standard for experiments above 300 GPa (Refs ), allowing one to study other materials at high pressure. The data on fused silica provide added information about silica in this regime. We present results of precision EOS measurements of the fused-silica Hugoniot from 200 to 1600 GPa using the impedance-matching technique. Previous work measured the fused-silica Hugoniot above the melt curve from 200 to 900 GPa (Refs. 23 and 25). This work extends measurements of the fused-silica Hugoniot well into the dissociated regime and provides lower-pressure data that agree with that of Qi et al. 25 The latter is significant in that the results of laser-driven experiments using -quartz as an impedance-matching standard agree well with those from direct-impact measurements using aluminum flyer plates driven by magnetic-field acceleration. This agreement increases confidence in the release model recently developed for silica (quartz) as a standard. 22 II. METHOD The Rankine Hugoniot equations relate the material conditions behind a shock discontinuity to those in front of it. 31 The relations for conservation of mass and momentum give 0 1 up U, (1) s 3

4 P 0Usup, (2) where is the density, 0 is the initial (unshocked) density, u p is the particle velocity, U s is the shock velocity, and P is the pressure. If P 0 and 0 are known, measurement of two quantities (typically U s and u p ) closes these equations and provides an EOS point on the material s Hugoniot. In this work, the Hugoniot of fused silica was determined by impedance matching (IM) to an -quartz standard. 22 The IM method relies on the Hugoniot and release curves of a standard to infer the sample s particle velocity from measurements of shock velocity in the standard and the sample. Quartz has been recently been established as the preferred standard for EOS experiments because of its transparency at ambient conditions and high shock reflectivity This technique used the quartz Hugoniot and release curves derived by Knudson and Desjarlais to perform the IM analysis. 18,22 These experiments comprise ten older, previously unpublished, experiments on the OMEGA laser (shots to and 64348) 32 and 18 recent experiments using the OMEGA EP laser, 33 both at the University of Rochester s Laboratory for Laser Energetics. Both of these frequency-tripled Nd:glass lasers operate at a wavelength of 351 nm. The shock pressures in these experiments were generated using laser pulses ranging from 2- to 6-ns duration, with intensities ranging from ~0.2 to W/cm 2. The laser spots were all smoothed using distributed phase plates 34 to achieve shocks that with planar regions of either 750 or 1100 m in diameter. 4

5 The targets consisted of 3-mm 3-mm flat, z-cut, -quartz baseplates with a nominal thickness of either 50 or 75 m. A 15- m-thick layer of parylene (CH) was deposited on the front of the target as a low-z ablator that enhanced the ablation pressure and reduced the production of hard x rays. The laser plasmas had temperatures of 1 to 2 kev(ref. 35) therefore, x rays generated from that plasma have an attenuation length <5 m in the quartz pusher. (Ref. 36) The shots where the intensity exceeded W/cm 2 were part of an older set of experiments with a 100 m- thick quartz baseplate and thicker ablator. For these experiments, 1-D hydrodynamic simulations using the hydrocode LILAC 37 gave peak electron temperatures in the ablation plasma of ~3 kev. Simulations also displayed negligible preheat after 50 m for the shots at W/cm 2 and 35 m for shots with intensities W/cm 2. This is consistent with earlier simulations by Brygoo et al. at intensities up to This limits the preheating of the quartz baseplate to regions far from the point where IM is performed. Adjacent fused-silica and -quartz samples were glued onto the back of the baseplate using an ultralow-viscosity, UV-cured epoxy that produced glue layers <3 m thick. A schematic of the target assembly is shown in Fig. 1(a). The fused-silica and quartz samples had initial densities of 2.20 g/cm 3 and 2.65 g/cm 3, respectively. The refractive indices at 532 nm were for the fused silica and for the quartz. Shock velocities were measured using a line-imaging velocity interferometer system for any reflector (VISAR) that measures the phase change caused by Doppler-shifted light reflecting off a moving surface. At pressures above 150 GPa, both the fused silica and quartz melt and form reflective shock fronts. 16 The materials are 5

6 both initially transparent so VISAR measures the velocity of the shock front within the material. Antireflective coatings were applied to the back surface of both samples to eliminate ghost reflections from that surface. To resolve 2 ambiguities, two VISAR s with different velocity sensitivities were used. These sensitivities were km/s/fringe and km/s/fringe for the quartz and km/s/fringe and km/s/fringe for the fused silica for shots using the OMEGA EP laser. The more-sensitive etalon for shots on the OMEGA laser gave km/s/fringe in the quartz and km/s/fringe in the fused silica. The VISAR images provided phase shifts resulting from changes in velocity that were analyzed using the Fourier transform method. 41 The error in determining the phase was estimated as ~3% of a fringe, resulting in velocity measurements with <1% precision because of multiple fringe jumps produced by the shock. The VISAR system used a frequency-doubled Nd: YAG laser operating at 532 nm, and the two streak cameras used 9- and 15-ns sweep durations. Figures 1(b) and 1(c) provide an example of the raw data and the extracted velocity profile. The experimental observables were the shock velocity in the quartz and fusedsilica samples at the point at which the shock transited the quartz/fused-silica interface. The shock velocities in the quartz and fused silica were measured immediately before and after the interface, respectively. High-precision impedance-match experiments require intimate contact between the pusher (in this case, quartz) and sample. However, target fabrication involves gluing the fused-silica sample to the quartz witness. This introduces a thin intermediate layer that has little effect on the shock propagation. Traditionally this is accounted for by fitting the shock velocity in the sample over a short (~300- to 500-ps) window and linearly extrapolating across the glue layer to determine the effective shock 6

7 velocity assuming intimate contact. 35 Inclusion of the quartz witness allows for a moreprecise treatment of the glue, by monitoring the shock velocity and planarity across the glue layer. In doing this, the exact change in shock velocity across the glue is determined because the velocity must be continuous between the quartz pusher and witness. For targets with the witness, the glue layer was treated as a perturbation to the quartz shock velocity and a zeroth-order unsteady wave correction was applied to determine the equivalent perturbation to the fused-silica shock velocity. 42 This technique measures the amplitude of the decay in shock velocity, Q U s, across the glue layer in the quartz and multiplies it by a factor FS Q G Us Us, where U FS s and Q U s are the average shock velocity in the fused silica and quartz, respectively, giving U FS Q s G U s. The averages are taken over identical temporal windows after the shock has transited the glue and entered the samples. The perturbation to the fused-silica shock velocity FS U s is scaled by the transit time across the glue layer on the fused-silica side relative to that on the quartz side and added to the shock velocity measured when the shock entered the fused-silica sample. This gives the expected shock velocity if no glue layer was present. The difference between the zeroth-order correction and linear extrapolation was approximately half the uncertainty in the shock velocity from VISAR, implying that this technique is consistent with the traditional method. For targets with no witness (shots and 64348), the traditional method of linearly extrapolating the shock velocity across the glue was used to determine the expected velocity at the theoretical quartz/fused-silica interface. 35 These targets used a thicker ablator which eliminated the 7

8 perturbations shown in Figures 1(b) and 1(c) and allowed for accurate fitting of the shock velocity after the interface. Errors were calculated using the Monte Carlo method, which generates normally distributed random variables based on the mean and standard deviation of an observed quantity. In these experiments, the random errors were the measured shock velocities in the quartz and fused silica and variances in the initial density of the samples, which were assumed to be negligible. For these velocities, this resulted in an observation error <1% from VISAR. Systematic errors that affected the measurement were the uncertainties in the cubic Us up Hugoniot fit for -quartz and the quartz release isentrope from Ref. 22. In the Monte Carlo simulations, a Cholesky decomposition was used to determine correlated normally distributed random variables from the covariance matrices for the quartz Hugoniot fit and effective Grüneisen parameter in the release model. The quartz shock state (P, ) was determined from the perturbed shock velocity and Hugoniot fit. The release curve was calculated from this point and the fused-silica shock state determined from the intersection of the quartz release curve and fused-silica Rayleigh line. This process was repeated 10 5 times and the error bars represent a one uncertainty in pressure and density. III. RESULTS/DISCUSSION A total of 28 impedance-matching points were used to determine the Hugoniot of fused silica, ranging in pressure from 260 GPa up to 1570 GPa. The results are given in Table I. The shock speed s dependence on particle velocity (U s u p curve), shown in Fig. 2(a), was fit to a cubic polynomial (red dashed line), and the exponential form 8

9 du Us a bup cupe p (blue line). The error bars on the data are of similar size as the points in the graph. The cubic and exponential fits were both made using orthogonally weighted 43 nonlinear-least-squares methods to the data in this article and that from both Qi et al. 25 and Lyzenga et al. 11 The coefficients and covariance matrix for the cubic fit are given in Tables II and III. Similarly, the parameters for the exponential model are given in Tables IV and V. The exponential fit using coefficients from Qi et al. 25 are also shown in Fig. 2(a) (dashed dotted line). The residuals for each of these fits were computed and, for clarity, a low-order spline was fit to each set of residuals [shown in Fig. 2(b)]. Linear (green dashed line) and quadratic (purple dotted line) fits are also shown for contrast. These residual plots show that the cubic fit (red dashed line) and our new exponential fit (blue line) replicate the data to similar precision. Importantly, the exponential fit given by Qi et al. (dashed dotted line) does not fit these data as well, predicting slightly stiffer behavior at higher pressures. This is demonstrated in the P Hugoniot data in Fig. 3 where the fit determined by Qi lies above that to the experimental results. It should be noted that the Qi fit was based on their molecular dynamics simulations that agreed with their data that reached only ~630 GPa. The simulations were performed for pressures of 86 to 1500 GPa and those simulation results were the basis for their exponential fit. Note that our data agrees very well with the data from Qi et al. However, at the higher pressures, the first-principles molecular dynamics (FPMD)-based exponential fit of Qi et al. is inconsistent with the data presented here. This can be seen in the residuals in Fig. 2(b) where the Qi model (dashed dotted line) deviates from the data at u p > ~12 (km/s). Below that velocity the models and data agree well. The deviation implies that the 9

10 FPMD results overpredict the stiffening of fused silica at high pressure. In the regime where the simulations are benchmarked by experimental results, the shocked fused silica reaches a compression ratio of = 2.6. These experiments and the simulations reach ~3.2-fold compression, where the Hugoniot curve is twice as steep as at = 2.6. The slope of the fit to FPMD results is ~10% greater than that to experimental results, which enables the difference between the curves to become significant when driving to increasing pressures. IV. CONCLUSION The Hugoniot of fused silica shocked to the fluid state was measured by impedance matching to -quartz for pressures up to 1600 GPa. The results from these laser-driven experiments are in good agreement over the range of 200 to 600 GPa with results using magnetically driven aluminum flyer plates a significant departure from past experience comparing the two drive platforms. The results extend the measurements for the fused-silica Hugoniot to 25% higher compression and approach maximum compressions reached for solids in planar geometry (~3.6).(Ref. 44) At these pressures, the difference in slope of ~10% between the FPMD determined fit of Qi et al. 25 and the experimental fit becomes significant. This results in the FPMD results predicting stiffer behavior in fused silica at high pressures. Using this data and that of Qi and Lyzenga, a new U s u p curve was generated for the range 200 to 1600 GPa. The cubic polynomial presented in Table II and a fit using the form dup Us a bup cupe given in Table IV were found to model the data equally well. The significant curvature in the cubic model is undesirable for extrapolation to pressures above 1600 GPa. Therefore, the exponential 10

11 model is preferred for extrapolation since at higher pressures it asymptotes to a linear U s u p relationship. ACKNOWLEDGMENT This material is based upon work supported by the Department of Energy National Nuclear Security Administration under Award Number DE-NA , the University of Rochester, and the New York State Energy Research and Development Authority. The support of DOE does not constitute an endorsement by DOE of the views expressed in this article. 11

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16 FIGURES CAPTIONS FIG. 1. (a) Target schematic showing the parylene ablator, quartz baseplate, quartz witness, and fused-silica sample. (b) Raw VISAR data for shot Target orientation is the same as in (a). The OMEGA EP line-imaging VISAR streak provides a temporal streak with 1-D spatial information of the shock velocity as the front propagates through the baseplate, thin glue layer, and quartz witness and fused-silica sample. The oscillations in shock velocity are the result of reverberations in the thin ablator with subsequent perturbations slightly increasing the average pressure as a function of time. (c) Extracted velocity for shot showing quartz (blue) and fused silica (red). FIG. 2. (a) U s u p plot for fused silica. Experimental results from this work (orange diamonds), Qi et al. 25 (blue squares), and Lyzenga et al. 11 (purple triangles) are shown. The exponential fit from Qi et al. (black dashed dotted line) and cubic (red dashed line) and exponential (solid blue line) fits from this work are also displayed. The error bars on the data are of similar size as the points in the graph. (b) Fits to residual of U s u p relations using a low-order smoothing spline. Cubic (red dashed line) and exponential (blue line) fits from this work agree with the data. Linear (green dashed line) and quadratic (purple dots) demonstrate systematic error in the fit, and exponential fit from Qi et al. (black dashed dotted line) diverges from experimental results for u p > 12 km/s (P > ~500 GPa). FIG. 3. P Hugoniot curve. Exponential fit from Qi et al. 25 clearly lies above the data and fits computed using this work and data from Qi et al. and Lyzenga et al. 11 This 16

17 implies that the fit by Qi is less compressible than the experimental data. Exponential fit and cubic fit from this work are indistinguishable for pressures <1400 GPa. Above 1400 GPa, the cubic fit begins to diverge and is not valid for extrapolation to pressures >1600 GPa. All data use the same labels as Fig. 2(a). Los Alamos Scientific Laboratory shock Hugoniot data (blue circles) and the SESAME 7386 table (purple line) are included for completeness. 17

18 Tables TABLE I. Shock parameters for fused silica determined from impedance matching to the FS FS quartz standard. U S, meas and U S, corr are the shock velocities measured at the fused- silica/glue interface and corrected for the glue layer using linear extrapolation (shots , 64348) and the zeroth-order correction (shots , ), respectively. Uncertainties in uncorrected and corrected shock velocities are identical. Shot# Q U S (km/s) FS U S,meas FS U S,corr FS u p FS P (GPa) (km/s) (km/s) (km/s) (g/cm 3 ) FS 18

19 TABLE II. Fit parameters for cubic U s u p relation of form U 2 3 s a 0 a 1 u p a 2 u p a 3 u p. a 0 km s 1 a a km s a 3 km s

20 2 a0 TABLE III. Covariance matrix elements for the cubic U s u p relation given in Table II. a a 0 1 a a 0 2 a a a1 a a 1 2 a a a2 a a 2 3 ( 10 2 ) ( 10 2 ) ( 10 3 ) ( 10 5 ) ( 10 3 ) ( 10 4 ) ( 10 5 ) ( 10 5 ) ( 10 6 ) ( 10 8 ) a3 20

21 TABLE IV. Fit parameters for exponential U s u p relation of dup s p p. form U a bu cu e a (km/s) b c d (km/s)

22 TABLE V. Covariance matrix elements for the exponential U s u p relation given in Table IV. 2 a ( 10 1 ) a b ( 10 2 ) a c ( 10 3 ) a d ( 10 2 ) 2 b ( 10 4 ) b c ( 10 4 ) b d ( 10 4 ) 2 c ( 10 2 ) c d ( 10 3 ) 2 d ( 10 3 )

23 (a) Quartz baseplate Parylene ablator Quartz witness OMEGA EP VISAR Fused silica (b) x (nm) t (ns) (c) Us (km/s) E24319J t (ns) 5 6 7

24 U s (km/s) Exponential (Qi et al.) Exponential (this work) Cubic (this work) This work Qi et al. Lyzenga 1983 (a) (b) Residual fit E24320J U p (km/s)

25 Pressure (GPa) Exponential (Qi et al.) Exponential (this work) Cubic fit This work Qi et al. LASL Hugoniot data Lyzenga SESAME 7386 E24321J t (g/cm 3 )